OpenTSTOOL User Manual Version 1.2 (2/2009) Christian Merkwirth Ulrich Parlitz Immo Wedekind David Engster Werner Lauterborn Drittes Physikalisches Institut Universit¨at G¨ottingen Contact: [email protected] 2 Contents 1 At a glance 7 2 Download and Installation 9 2.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Installation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Global installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Deinstalling TSTOOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 First Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Compiling mex files 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Problems with compiling mex files . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Copyright notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 3 First Steps 3.1 15 Example analysis of a time-series from a chaotic Colpitts oscillator . . . . . . . . . . . 4 Nearest Neighbors Searching 15 19 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Approximate nearest neighbors searching . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Range searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Matlab mex-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4.1 nn prepare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4.2 nn search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4.3 range search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5 Example session 5 Handling the Graphical User Interface 25 5.1 Filelist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3.1 26 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5.3.2 Methods I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3.3 Methods II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3.4 Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3.5 Modify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.6 Macro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3.7 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3.8 Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3.9 View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6 Mex-Function Reference 33 6.1 akimaspline - Cubic spline interpolation using Akima splines . . . . . . . . . . . . . . 33 6.2 amutual - compute auto mutual information function . . . . . . . . . . . . . . . . . . 34 6.3 baker - Generate Baker time-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.4 boxcount - Classical boxcounting algorithm . . . . . . . . . . . . . . . . . . . . . . . . 35 6.5 cao - Determine minimum embedding dimension by Cao’s method . . . . . . . . . . . 35 6.6 chaosys - integrate dynamical system given by a set of ordinary differential equations 36 6.7 corrsum - Computation of the correlation sum . . . . . . . . . . . . . . . . . . . . . . 38 6.8 corrsum2 - Computation of the correlation sum . . . . . . . . . . . . . . . . . . . . . . 39 6.9 fnearneigh - Fast nearest neighbor search . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.10 gendimest - Estimate generalized dimension spectrum . . . . . . . . . . . . . . . . . . 41 6.11 henon - Generate henon time-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.12 largelyap - Compute separation of nearby trajectories . . . . . . . . . . . . . . . . . 42 6.13 nn prepare - Do nearest neighbor preprocessing . . . . . . . . . . . . . . . . . . . . . 43 6.14 nn search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.15 predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.16 range search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.17 return time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.18 takens estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.19 tentmap - Generate tentmap time-series . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.20 Class signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.20.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.20.2 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.20.3 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.21 Class description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.21.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.21.2 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.21.3 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.22 Class core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 6.22.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.22.2 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.22.3 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.23 Class achse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.23.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.23.2 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.23.3 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.24 Class unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.24.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.24.2 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.24.3 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.25 Class list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.25.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.25.2 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.25.3 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7 Frequently asked questions 93 7.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2.1 Introduction and general information . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2.2 Installation of TSTOOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2.3 Working with TSTOOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.2.4 Extending TSTOOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2.5 Miscellaneous questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.2.6 Frequently encountered errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5 Chapter 1 At a glance What is TSTOOL ? TSTOOL is a software package for signal processing with emphasis on nonlinear time-series analysis. Objectives • Implement existing algorithms for nonlinear time-series analysis • Develop new methods for specific data analysis problems • Create an expandable platform for signal processing Implementation The package is written partly in Matlab and partly in C++. Advantages of Matlab are : • Reduced code development time • Extensive collection of intrinsic mathmatical functions • Excellent graphic capabilities • High portability from Unix to Windows NT and other platforms C++ is used for computationally demanding algorithms. Graphical user interface A graphical user interface (GUI) gives access to the underlying signal processing commands. Parameters for the commands are set via dialog windows. 7 Chapter 2 Download and Installation 2.1 Installation Unpack the compressed TSTOOL distribution into a directory, e.g C:\Program Files on Windows, /usr/local on Unix. This can be done with an unpacking tool like Winzip if you are working with Windows, or gzip -dc filename.tgz | tar -xvf - if you are working with Unix. After unpacking you will get a new directory named OpenTSTOOL which should now contain: • tstoolInit.m - script that calls settspath.m if necessary • settspath.m - script that does path settings • tstoolbox - the directory that contains all TSTOOL functions, compiled mex files etc. • mex-dev - Source code of the C++ parts of TSTOOL • Doc - HTML/PDF Documentation • info.xml, tstoolicon.gif - Files for the Matlab start menu • gpl.txt - Gnu General Public License Important note for Windows users: If you want to use the pre-compiled mex-files which ship with OpenTSTOOL, you will probably have to install the Visual C++ 2008 run-time libraries, which can be downloaded from the Microsoft web site. As of the writing of this document, the URLs for these packages are http://www.microsoft.com/downloads/details.aspx?FamilyID=9b2da534-3e03-4391-8a4d-074b9f2bc1bf for 32bit and http://www.microsoft.com/downloads/details.aspx?familyid=bd2a6171-e2d6-4230-b809-9a8d7548c1b6 for 64bit. If these URLs are no longer valid, just search for “Microsoft Visual C++ 2008 Redistributable Package” and “x86” or “x64” for 32bit and 64bit, respectively. If you do not have these run-time libraries installed, you will get the following error message when trying to run a mex-file: “This application has failed to start because the application configuration is incorrect.” 9 2.1.1 Installation methods The simplest way to invoke OpenTSTOOL is to start Matlab and change to the OpenTSTOOL directory (using the GUI or the cd command). Once you have entered that directory, you should see the OpenTSTOOL toolbox in your Matlab Start menu under the item Toolboxes, and you can start the graphical user interface from there. Note that TSTOOL is not really installed yet, so that after restarting Matlab you will have to enter the OpenTSTOOL directory again before you will be able to use the toolbox. If you want to install OpenTSTOOL permanently, the directories from the toolbox must be in the Matlab path. There are several methods to include these directories, depending on you having full control over the Matlab installation or using a network wide installation as a normal user with reduced privileges. • If you have full control over the installation: Enter the OpenTSTOOL directory and choose the item Installation under Toolboxes → OpenTSTool in the Matlab Start menu. It will install the OpenTSTool directories in the Matlab search path and call the path tool. Now simply click save to store the current path for future sessions. • If you are working with a networked installation: Enter the path tool as just described in the previous paragraph, but after you click on save, you will get an error which states that the path could not be saved. Now, choose Yes to save the file pathdef.m in your userpath, which you can see by calling the command userpath in the Matlab shell (on UNIX this is usually the directory <HOMEDIR>/matlab or <HOMEDIR>/Documents/matlab. Create it if it does not yet exist). You might have to set the environment variable MATLAB USE USERPATH to 1 to make this work. • If don’t have a graphical display and are working with the shell : Enter the OpenTSTOOL directory, call tstoolInit and then savepath. • Another possibility is to create a file startup.m in your userpath (usually <HOMEDIR>/matlab or <HOMEDIR>/Documents/matlab), containing the following lines: addpath(’<full path to the OpenTSTOOL-Dir>’); settspath(’<full path to the OpenTSTOOL-Dir>’); Again, on UNIX systems you might have to set the environment variable MATLAB USE USERPATH to 1 to make this work. 2.1.2 Global installation Last but not least, if all users of a network wide matlab installation should have access to TSTOOL, just insert the paths which are set by the script settspath in the global path from Matlab. 2.1.3 Deinstalling TSTOOL If you want to deinstall TSTOOL, simply remove the OpenTSTOOL directory, start pathtool in Matlab and remove every directory containing OpenTSTOOL. Then save the path for future sessions. 2.2 First Steps 1. Start Matlab 2. Install paths as described in 2.1.1. 3. Enter tsdemo on the Matlab console. This should start a short demo script. 10 4. Enter tsdemo2 on the Matlab console. This should run a second script that shows an analysis of a chaotic signal. 5. Note that you can also start the demos through the Matlab Start menu. 6. Enter opentstool to start the graphical user interface for the TSTOOL package. 7. You can also access the manual any time through the OpenTSTOOL entry in the Matlab start menu. 2.3 Compiling mex files If you do not find appropriate pre-compiled mex files for your platform on our home page, you can compile them yourself. You will need an installed C++ compiler for doing this; the C compiler that ships with MATLAB (LCC) does not compile C++ code at all and will therefore not work. • GNU/Linux : The free compiler suite gcc usually comes pre-installed with most distributions. You will need at least version 3.4 or newer (check with gcc -v). Run mex -setup in MATLAB and choose the gccopts.sh options file for compilation. • Solaris: You can also use gcc, but the C++ compiler from Sun Forte Studio should also work (Version 7 or newer). For using the latter, make sure the command cc runs the correct compiler in the shell, call mex -setup in MATLAB and choose the system’s ANSI compiler. • Mac OS X : The gcc suite is available through Apple’s XCode Developer Tools, which can be downloaded from http://developer.apple.com/TOOLS/xcode (registration required). As for GNU/Linux, run mex -setup and choose gccopts.sh. • Windows: Unfortunately, compiling mex files under Windows ist more difficult to do, since each MATLAB version is pretty picky about which version of which compiler it supports. If you happen to have Microsoft Visual C++ Professional, this should usually be detected by mex -setup and it should compile the C++ files just fine. If you want to use the Express edition, which you can download free of charge from Microsoft, usually only the newest MATLAB version (currently 7.7) will support it directly. For older MATLAB versions, you will also have to install the Microsoft Windows SDK for Windows Server 2008 and .NET Framework and copy some files in the right places. Fortunately, there’s an tutorial for this, which you’ll find at http://www.mathworks.com/matlabcentral/fileexchange/22689 Alternatively, like on Unix systems, you can install the free gcc compiler on Windows, which is available under the name MinGW (minimalist gcc for Windows). Unfortunately, Matlab does not officially support this compiler under Windows, so you cannot use it by simply calling mex -setup. Instead, you can try to use the free tool GnuMex to create a proper compilation batch file for you. Please visit the following web sites for details and downloads: GnuMex: http://gnumex.sourceforge.net MinGW: http://www.mingw.org Note that GnuMex will currently not work on 64bit Windows systems out of the box! After you have set up the mex tool, enter the subdirectory mex-dev and call makemex. This function will compile all the necessary files and copy them into the tstoolbox/mex/<MEXEXT> subdirectory. 11 2.3.1 Problems with compiling mex files Dependent on the version you are using, Matlab will support only a certain range of gcc compilers. If you are using an unsupported gcc version, you will see a warning message like this during compilation: Warning: You are using gcc version "X.X.X". The earliest gcc version supported with mex is "Y.Y.Y". The latest version tested for use with mex is "Z.Z.Z". To download a different version of gcc, visit http://gcc.gnu.org While compiling with a newer gcc version often works despite the warning, you should still try to use a gcc version which your version of Matlab supports. On many Unix systems (especially most Linux distributions), more than one version of gcc is installed, and often you can use one certain version by using an explicit command like gcc-X.Y. You can put the appropriate command into the mexopts.sh file (instead of just gcc and g++), which Matlab created for you during mex -setup. On Unix systems, it can be found in the directory .matlab/VERSION under your home directory. Another problem can occur due to Matlab coming with its own versions of the standard C/C++ and gcc libraries (at least under Unix systems). This can lead to problems if your mex files were linked against another C library installed on your system. You will typically get an error like the following when calling such a mex file: <MATLABROOT>/sys/os/glnx86/libgcc_s.so.1: version ‘GCC_4.2.0’ not found (required by /usr/lib/libstdc++.so.6) There exist basically two options for solving this problem. Note that as far as we know, both are not officially supported by The Mathworks, but usually work nonetheless. • First option: move the files libstdc++*, libgcc* and libg2c* from <MATLABROOT>/sys/os/<SYSTEM> to another location, for example a subdirectory named old. You will thereby enforce the libraries installed on your system. If you encounter problem afterwards, simply move the libraries back and restart Matlab. Of course, you need control over the Matlab installation for doing this. • Second option: Set the environment variable LD PRELOAD to enforce usage of your installed system libraries. For example, you may call Matlab the following way from the command line: LD_PRELOAD=/usr/lib/libstdc++.so.6:/lib/libgcc_s.so.1 matlab This option is a bit more cumbersome, but has the advantage that you can do this as normal user. Note that this command is only an example; check your library directories for the correct paths and file names. 2.4 Pitfalls See also the FAQ (Frequently Asked Questions) 7! 1. As of TSTOOL version 1.2, the main function for creating the TSTOOL GUI is called opentstool instead of tstool to avoid collision with the time series toolbox from The Mathworks. 2. When using Winzip, enable Use path information to make sure that subdirectories are created. 3. TSTOOL will not work with Matlab version prior to 6.5! 4. It’s not a good idea to place the TSTOOL distribution into the Matlab directory. We obtained reports about strange bugs occuring when the TSTOOL distribution is extracted into the directory where the Matlab system is installed. 12 2.5 Copyright notice TSTOOL falls unter the GNU General Public License. See gpl.txt in the OpenTSTOOL directory or http://www.physik3.gwdg.de/tstool/gpl.txt. . 13 Chapter 3 First Steps 3.1 Example analysis of a time-series from a chaotic Colpitts oscillator In this section we briefly demonstrate basic steps for anaysing a chaotic time series. The methods used will be explained ind maore detail in the following sections. >> s = signal(’colpitts.dat’,’ascii’) s = signal object Dlens : 6001 X-Axis 1 : | Name : colpitts Type : Attributes of data values : | Comment : History : 17-Aug-1999 15:08:24 : Imported from ASCII file ’colpitts.dat’ By entering the above command line, the overloaded constructor for class signal was called. Giving a filename as argument tells the constructor to load the datafile and convert it into a signal object. The datafile ’colpitts.dat’ contains a time-series generated by an electronical Colpitts circuit that oscillates chaotically. To plot signal s, just issue the following command : view(s); 15 Lets find a good choice for a delay-time by using the first minimum of the auto mutual information function a = amutual(s,32); view(a); the first minimum of the auto mutual information can be found at four. Now we need to know the minimal embedding dimension for the colpitts signal. We use Cao’s method with a delay time of four, a maximal dimension of eight, three nearest neighbors and 1000 reference points. c = cao(s,8,4,3,1000); view(c); There’s a kink in the graph produced by Cao’s method at three. So now do a time-delay reconstruction of the Colpitts signal with embedding dimension 3 and delay 4. 16 e = embed(s, 3, 4); view(e); What’s the correlation dimension of the reconstructed data set ? First let’s take a look at the scaling of the correlation sum versus the radius (as log-log plot). view(corrsum(e, -1, 0.05, 40, 32)); Next, we use the Takens estimator for the correlation dimension. It needs basically the same input arguments as the function corrdim2. >> takens_estimator(e, -1, 0.05, 40) ans = 1.9483 And what about it’s largest Lyapunov exponent ? To estimate the largest Lyapunov exponent, we take a look at the scaling of the prediction error. view(largelyap(e, 1000, 300, 40, 2)); 17 18 Chapter 4 Nearest Neighbors Searching An integral part of a majority of methods for nonlinear time series analysis is searching for nearest neighbors. The perfomance of these methods depends strongly of the perfomance of the employed nearest neighbor algorithm. Thus, choosing an efficient nearest neighbor algorithm should be done very carefully. 4.1 Definition Definition : A set P of data points in D-dimensional space is given. Then we define the nearest neighbor to some reference point q (also called query point) to be the point of data set P that has the smallest distance to q (we don’t issue the problem of ambiguity at this point).The more general task of finding more than one nearest neighbor is called k nearest neighbors problem. In general, the reference point q is an arbitrarily located point, but it is also possible that q is itself a member of data set P (as illustrated in the figure, where five neighbors to q (excluding self match) are found). 4.2 Approximate nearest neighbors searching Approximate nearest neighbors algorithms report neighbors to the query point q with distances possibly greater than the true nearest neighbors distances. The maximal allowed relative error epsilon is given as a parameter to the algorithm. For epsilon=0, the approximate search returns the true (exact) nearest neighbor(s). Computing exact nearest neighbors for data set with fractal dimension much higher than 6 seems to be a very time-consuming task. Few algorithms seem to perform significantly better than a bruteforce computation of all distances. However, it has been shown that by computing nearest neighbors approximately, it is possible to achieve significantly faster execution times with relatively small actual errors in the reported distances. 19 4.3 Range searching In the task of range searching, we ask for all points of data set P that have distance r or less from the query point q. Sometimes range searching is called a fixed size approach, while k nearest neighbors searching is called a fixed mass approach. 4.4 Matlab mex-functions 4.4.1 nn prepare nn prepare does the preprocessing for a given data set pointset. The returned data structure atria contains preprocessing information that is necessary to use nn search or range search. Preprocessing and searching is divided into different mex-files to give the user the possibility to reuse the preprocessing data (contained in atria) when doing multiple searches on the same point set. However, as soon as the underlying point set is changed or modified, one has to recompute atria for the changed point set. Syntax: • atria = nn_prepare(pointset) • atria = nn_prepare(pointset, metric) • atria = nn_prepare(pointset, metric, clustersize) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • metric - (optional) either ’euclidian’ or ’maximum’ (default is ’euclidian’) • clustersize - (optional) threshold for clustering algorithm, defaults to 64 4.4.2 nn search nn search does exact or approximate k-nearest neighbor queries to one or more query points. These query points can be given explicitly or taken from the data set of points (see below). Before one can use nn search, one has to call nn prepare to compute the preprocessing information. However, as long as the input point set isn’t modified, the preprocessing information is valid and can be re-used for multiple calls to nn search or range search. Syntax: 20 • [index, distance] = nn_search(pointset, atria, query_points, k) • [index, distance] = nn_search(pointset, atria, query_points, k, epsilon) • [index, distance] = nn_search(pointset, atria, query_indices, k, exclude) • [index, distance] = nn_search(pointset, atria, query_indices, k, exclude, epsilon) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • atria - output of nn prepare for pointset • query points - a R by D double matrix containing the coordinates of the query points, organized as R points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • k - number of nearest neighbors to compute • epsilon - (optional) relative error for approximate nearest neighbors queries, defaults to 0 (= exact search) • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • index - a matrix of size R by k which contains the indices of the nearest neighbors. Each row of index contains k indices of the nearest neighbors to the corresponding query point. • distance - a matrix of size R by k which contains the distances of the nearest neighbors to the corresponding query points, sorted in increasing order. 4.4.3 range search The routine range search does a range search to one or more query points. These query points can be given explicitly or taken from the data set of points (see below). Before one can use range search, one has to call nn prepare to compute the preprocessing information. However, as long as the input point set isn’t modified, the preprocessing information is valid and can be re-used for multiple calls to nn search or range search. Syntax: • [count, neighbors] = range_search(pointset, atria, query_points, r) • [count, neighbors] = range_search(pointset, atria, query_indices, r, exclude) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D 21 • query points - a R by D double matrix containing the coordinates of the query points, organized as R points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points • r - range or search radius (r > 0) • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • count - a vector of length R contains the number of points within distance r to the corresponding query point • neighbors - a Matlab cell structure of size R by 2 which contains vectors of indices and vectors of distances to the neighbors for each given query point. This output argument can not be stored in a standard Matlab matrix because the number of neighbors within distance r is not the same for all query points. The vectors if indices and distances for one query point have exactly the length that is given in count. The values in the distances vectors are not sorted. 4.5 Example session % create a 3-dimensional data set with 100000 points pointset = rand(100000, 3); % do the preprocessing for this point set atria = nn_prepare(pointset, ’euclidian’); % now search for 2 (exact) nearest neighbors, using points 1 to % 10 as query points, excluding self-matches [index, distance] = nn_search(pointset, atria, 1:10, 2, 0) index = 5618 38209 54991 38429 4114 72121 13678 26022 86042 24830 96574 84549 60397 59732 76991 452 59332 16718 38436 44434 distance = 22 0.0101 0.0078 0.0132 0.0050 0.0087 0.0124 0.0129 0.0046 0.0101 0.0156 0.0175 0.0134 0.0167 0.0223 0.0097 0.0189 0.0168 0.0110 0.0103 0.0177 % now do a range search for radius 0.0224, using points 1 to 10 as % query points, excluding self-matches [count, neighbors] = range_search(pointset, atria, 1:10, 0.0224, 0) count = 4 10 7 2 5 6 2 4 7 5 neighbors = [1x4 [1x10 [1x7 [1x2 [1x5 [1x6 [1x2 [1x4 [1x7 [1x5 double] double] double] double] double] double] double] double] double] double] [1x4 [1x10 [1x7 [1x2 [1x5 [1x6 [1x2 [1x4 [1x7 [1x5 double] double] double] double] double] double] double] double] double] double] % let’s see the indices of the points that are within range to the first query point neighbors{1,1} ans = 56921 97100 96574 5618 % let’s see the corresponding distances of the points that are % within range to the first query point neighbors{1,2} 23 ans = 0.0176 0.0186 0.0175 0.0101 24 Chapter 5 Handling the Graphical User Interface With the Graphical User Interface (GUI) most of TSTOOLs methods are available without coping with the command line syntax of every function. To invoke the GUI simply type opentstool at the matlab command prompt. If you invoke the GUI for the first time, you get informed that the GUI will generate a directory for temporary files. This will reside at OpenTSTOOL/datafiles on Windows systems and at ~/.tstool on Unix systems. First of all, how does the GUI looks like? There are three parts: • Filelist • Figure • Menubar 5.1 Filelist Every loaded or generated signal shows its name in an own line, they arranged hierarchicaly. The data of the signals is stored in seperated files in the directory generated at the first run. To process a special method, click on one of the signals in the filelist an choose the method from the menubar. 25 5.2 Figure This figure can show you the signal you have choosen from the filelist. Normaly this feature is switched off because of the time consumption especialy for large signals. To switch on this feature choose the menu item Option-Instant View-Small Window. 5.3 Menus 5.3.1 Signal All menu items in this menu have something to do with the filelist and the storage of the signals. • Load This menu item simply loads data into the GUI (more precise: load a signal and save it to the temporary data directory). A line in the fileview will be added with its filename. • Save Write the marked signal in the fileview to disk. • Import file from Generate a signal from foreign formats like ASCII, Matlab Vector, Soundfiles etc. See signal class constructor reference (6.20.3.66) for more information. • Export file from Write the marked signal in the fileview to disk in a foreign format. See signal/write reference (6.20.3.83) for more information. • Generate This menu item can generate signals. See mex/chaosys reference (6.6). • Audio playback Plays a scalar signal as audio with the matlab function soundsc. If there is no sampling rate set in the signal 8KHz will be used. • Rescan Normally all files with the correct extension in the temporary directory are displayed in the filelist. For some reason it is possible that a signal is displayed in the filelist but doesnt exist (e. g. an other process deleted this file) or some files missing in the filelist (e. g. if you simply copy a signal file in the temporary directory without using the Load menu item). In such situation use the Rescan menu item to let TSTOOL look up all files again correctly. This can take some time on slow machines or large signal files. Simply restarting the GUI will not make a rescan! • Remove entry If you want to remove a filelist entry use this menu item or simply type Ctrl-d. The selected filelist entry will disappear and the corresponding temporary file will be deleted. • Show A signal stores many information about the data. This information can be displayed by this menu item. • Edit – Desired type of plot Here you can choose the type of plot TSTOOL should use for your signal. See signal/view reference (6.20.3.82) for additional information. 26 – Descriptive parameters – Axes labels – Comment text 5.3.2 Methods I In this menu all methods with scalar input are grouped. Most of this methods invokes the underlying TSTOOL-function directly and do not need addition explanations. To enter the parameters a dialog box will be opened. For some parameters there is a checkbox at the right side named ’in units’. Normally this parameter is entered in the units of samples. When you switch on the checkbox you can also enter this parameters in units of the first axis. • Reconstruction – Time-Delay vectors see signal/embed (6.20.3.21) – Minimum Embedding Dimension (Cao’s) see signal/cao (6.20.3.9) • Spectral – FFT see signal/fft (6.20.3.22) – Periodogram see signal/spec (6.20.3.68) – Spectrogram see signal/spec2 (6.20.3.69) – Scalogram see signal/scalogram (6.20.3.62) • Derivative/Integration – Integrate see signal/int (6.20.3.33) – Differentiate see signal/diff (6.20.3.18) • Correlation and more – Auto Correlation see signal/acf (6.20.3.2) – Auto Mutual Information see signal/amutual (6.20.3.4) • Filter – Moving Average see signal/movav (6.20.3.44) – Median Filter see signal/medianfilt (6.20.3.40) – Multiresolution Analysis see signal/mutlires (6.20.3.45) 27 • Surrogate Data Generation – Permutation of Samples see signal/surrogate3 (6.20.3.74) – Theiler Alg. I see signal/surrogate1 (6.20.3.72) – Theiler Alg. II see signal/surrogate2 (6.20.3.73) • Surrogate Data Test – Time Reversibility see signal/trev (6.20.3.80) – Higher order moments see signal/tc3 (6.20.3.78) – Function see signal/surrogate test (6.20.3.75) • Prediction – Local Constant see signal/predict2 (6.20.3.55) • Misc – Squared Magnitude see signal/power (6.20.3.53) – Absolute Value see signal/abs (6.20.3.1) – Decibel Value see signal/db (6.20.3.16) – Histogram see signal/histo (6.20.3.30) 5.3.3 Methods II In this menu all methods with multivariate input signals are grouped. • Decompositions – PCA (Karhunen-Loeve) see signal/pca (6.20.3.49) – Archetypal Analysis see signal/arch (6.20.3.7) • Lyapunov Exponents – Largest see signal/largelyap (6.20.3.36) • Fractal Dimensions – Box Counting Approach 28 ∗ Capacity Dimension D0 see signal/boxdim (6.20.3.8) ∗ Information dimension D1 see signal/infodim (6.20.3.31) ∗ Correlation dimension D2 see signal/corrdim (6.20.3.11) – Correlation Sum Approach ∗ Correlation Sum D2 (GPA like approach) see signal/corrsum (6.20.3.12) ∗ Correlation dimension D2 (fixed number of pairs) see signal/corrsum2 (6.20.3.13) ∗ Takens Estimator D2 see signal/takens estimator (6.20.3.77) – Nearest Neighbor Algorithms ∗ Information dimension D1 (NNK) see signal/infodim2 (6.20.3.32) ∗ Fractal dimension spectrum see signal/fracdims (6.20.3.27) • Periodicity – Return Times see signal/return time (6.20.3.58) – Reciprocal local density see signal/localdensity (6.20.3.38) • Modeling – Polynom selection see util/pauswahl • Poincare Section see signal/poincare (6.20.3.52) • Prediction – Local Constant see signal/predict (6.20.3.54) 5.3.4 Utilities Some useful information about the signals can be retrieved by functions in this menu. • Minimum see signal/min (6.20.3.42) • Maximum see signal/max (6.20.3.39) • First Local Minimum see signal/firstmin (6.20.3.25) • First Local Maximum see signal/firstmax (6.20.3.24) 29 • First Zero Crossing see signal/firstzero (6.20.3.26) • Mean see mean (Matlab reference) • Standard Deviation see std (Matlab reference) • RMS root mean square • Compare two Signals Only the data values are compared. See core/compare (6.22.3.3) 5.3.5 Modify • Cut see signal/cut (6.20.3.15) • Swap Dimensions see signal/swap (6.20.3.76) • Reverse see signal/reverse (6.20.3.59) • Interpolations – Cubic Spline see signal/upsample (6.20.3.81) – Akima Spline see signal/upsample (6.20.3.81) – FFT Based see signal/upsample (6.20.3.81) • Normalize – Center around Zero see signal/center (6.20.3.10) – Scale by Factor see signal/scale (6.20.3.61) – Fit to Interval see signal/norm1 (6.20.3.47) – Center and Divide by STD see signal/norm2 (6.20.3.48) – Remove Trend see signal/trend (6.20.3.79) – Transform to Rang Values see signal/rang (6.20.3.56) • Split Multichannel Signal Splits up a n-channel signal in n signals by using signal/cut (6.20.3.15). 30 • Add two Signals see signal/plus (6.20.3.51) • Difference of two Signals see signal/minus (6.20.3.43) • Merge two Signals see signal/merge (6.20.3.41) 5.3.6 Macro TSTOOL records the processed commands for every signal. So TSTOOL knows how this signal is modified and can generate a matlab script with processes the same commands to arbitrary signal. The generated scripts will be saved in the directory scripts inside the directory for temporary files. • Create Macro from Signal Generate script named macro.m. • Show/Edit Macro • Rename Macro Renamed macros will be displayed at the end of this menu after restart of TSTOOL. • Apply Macro to Signal Invoke macro.m with the selected signal. • Apply Macro to all Invoke macro.m with every loaded signal. After the seperation line a list of all *.m-Files in the directory scripts is shown. Selecting one of these menus apply the corresponding script to the actual selected signal. 5.3.7 Options Here some settings can be edited. All these settings will be saved in the file tstool.mat in the temporary data directory. • Parameters – Reconstruction Parameters The default settings for the Reconstruction dialog box can be edited here (see menu Methods I - Reconstruction - Time Delay Vectors 5.3.2) – Default Window Type Used by FFT etc. • File and Directory Options The actual search directory for Load and Save can be edited here. Also the default file extension can be altered. • Instant View – Small Window If you switch on this feature, the small figure at the right of the filelist will show you the selected signal immediatly after selection. – Large Window Every time a new signal is generate (e.g. applying a command to an existing signal) e new figure window will be opened displaying it. 31 5.3.8 Help The menu Usage will start your web browser with the HTML-Version of this manual shipped with the OpenTSTOOL distribution. Run ’help docopt’ at the matlab command prompt to configure your web brower correctly. After the seperation line you can view the matlab command line help for every method of the signal class. The same information is in the HTML- and in the PDF-Version of the manual in class reference section 6.20. 5.3.9 View This menu invokes a new figure window viewing the selected signal using the signal/view command (6.20.3.82). You can also simply type Ctrl-v. 32 Chapter 6 Mex-Function Reference Parts of TSTOOL’s functionality are coded in mex-files. All TSTOOL mex-files are located in the directory tstoolbox/mex. It is possible to use these mex-files independently of the full TSTOOL installation. 6.1 akimaspline - Cubic spline interpolation using Akima splines Compared to Matlab’s built-in cubic spline, Akima spline interpolation better copes with discontinuities in a time series. Syntax: • yy = akimaspline(x,y,xx) Input arguments: • x, y - vectors describing knot data, see Matlab’s original spline function • xx - vector of positions at which the spline is evaluated Output arguments: • yy - evaluated function values Example: x = 1:100; y = floor((x + rand(1, 100))/ pi); xx = 1:0.1:100; yy = akimaspline(x, y, xx); plot(x,y, xx, yy, ’r’) 33 6.2 amutual - compute auto mutual information function Fast, but crude auto mutual information of a scalar timeseries for the timelags from zero to maxtau. The input time series should be much longer than maximal timelag maxtau. The algorithm uses equidistant histogram boxes, so results are bad in a mathematical sense. However, a fast algorithm based on ternary search trees to store only nonempty boxes is used. Syntax: • a = amutual(ts, maxtau, partitions) Input arguments: • ts - vector holding time series data • maxtau - maximal time lag • partitions - number of partitions for the one-dimensional histogram Output arguments: • a - vector of length maxtau+1, holding auto mutual information 6.3 baker - Generate Baker time-series Generate time-series from the iterated Baker map [150]. Syntax: • x = baker(length, [eta l1 l2 x0 y0]) Input arguments: • length - number of samples to generate • [eta l1 l2 x0 y0] - vector of parameters and initial conditions Output arguments: • x - time series Example: x = baker(2000, [0.6 0.25 0.4 rand(1,1) rand(1,1)]); plot(x(1:end-1,2), x(2:end,2), ’.’) 34 6.4 boxcount - Classical boxcounting algorithm boxcount is a fast algorithm that partitions a data set of points into equally spaced and sized boxes. The algorithm is based on Robert Sedgewick’s Ternary Search Trees [149] which offer a fast and efficient way to create and search a multidimensional histogram. Empty boxes require no storage space, therefore the maximum number of boxes (and memory) used can not exceed the number of points in the data set, regardless of the data set’s dimension and the number of partitions per axis. During processing, data values are scaled to be within the range [0,1]. All columns of the input matrix are scaled by the same factor, so no skewing is introduced into the point set. Syntax: • [a,b,c] = boxcount(point set, partitions) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D. D is limited to 128. • partitions - number of partitions per axis, limited to 16384. For convenience, if a vector is given, boxcount will iterate over all values of this vector. Output arguments: • a - vector of size D with: log2(sum(Number of nonempty boxes)) • b - vector of size D with: sum(p * log2(p)) , where p is the relative frequency of points falling into a box • c - vector of size D with: log2(sum(p*p)), where p is the relative frequency of points falling into a box Example: p = rand(50000, 4); p = p - min(min(p)); p = p ./ max(max(p)); [a,b,c] = boxcount(p, 16) 6.5 cao - Determine minimum embedding dimension by Cao’s method This mex-file applies Cao’s method [38] to the input data set. If the data set contains points of dimension D, it computes E and E* for a data set of dimension 1 (taken from the first column of the input data set), then for a data set of dimension 2 (taken from the first two columns) up to a dimension of D. Optionally, this algorithm extends Cao’s method in a straightforward manner to use more than one nearest neighbors. Syntax: • [E, E*] = cao(pointset, query indices, k) Input arguments: 35 • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • k - number of nearest neighbors to compute. Cao’s method can be extended to use more than only the first nearest neighbor (k=1). Output arguments: • E and E* are vectors of size D. Please refer the Cao’s article [38] for a precise description of their meaning. 6.6 chaosys - integrate dynamical system given by a set of ordinary differential equations chaosys gives the user the possibility to compute time series data for a couple of dynamical systems, among which are Lorenz, Chua, Roessler etc. This routine is not meant as a replacement for Matlab’s suite of functions for solving ODEs, but as a fast way to generate some data sets to evaluate the processing capabilities of TSTOOL. The integration is done by an ODE solver using an Adams Pece scheme with local extrapolation [151]. It is at least faster than Matlab’s native ODE solver. However, it is not possible to extend the set of systems without recompiling chaosys. Syntax: • x = chaosys(length, stepwidth, initial conditions, mode, parameters) Input arguments: • length - number of samples to generate • stepwidth - integration step size • initial conditions - vector of initial conditions • mode: – 0: Lorenz – 1: Generalized Chua : Double Scroll – 2: Generalized Chua : Five Scroll – 3: Duffing – 4: Roessler – 5: Toda Oscillator – 6: Van der Pol Oscillator – 7: Pendulum For an exact definition of the ODE systems, please refer to this header file. • parameters - vector of systems parameters. The order of the parameters is exactly the same as in the constructors of the DGL subclasses in the above file. Output arguments: 36 • x contains the output of the integration, organized as matrix of size samples by dim, where dim is the number of ODEs that define the system Example: x = chaosys(20000, 0.025, [0.1 -0.1 0.02], 0); plot(x(:,1)); Definitions of the ODEs: The parameters of the odes are a vector of [a,b,...]. Lorenz: dy1 = a(y1 − y2 ) dt dy2 = by1 − y2 − y1 y3 dt dy3 = y1 y2 + cy3 dt Generalized Chua: dy1 = a(y1 − by2 ) dt dy2 = by1 − y2 + y3 dt dy3 = −cy2 dt Duffing: dy1 = y2 dt dy2 = −y1 − y13 − by2 + a cos y3 dt dy3 =c dt R¨ ossler: dy1 = −y2 − y3 dt dy2 = −y1 + ay2 dt dy3 = b + y3 (y1 − c) dt Toda oscillator: dy1 = 1 + a sin(bt) − by2 − exp y1 dt dy2 = y1 dt van der Pol oscillator: dy1 = a sin(bt) − c(by22 − 1) − d2 y1 dt dy2 = y1 dt 37 pendulum: dy1 = a sin(bt) − cby2 − d sin y1 dt dy2 = y1 dt 6.7 corrsum - Computation of the correlation sum The topics correlation sum and correlation dimension estimation can also be found here. Syntax: • [c, d] = corrsum(pointset, query indices, range, exclude) • [c, d] = corrsum(pointset, query indices, range, exclude, bins) • [c, d] = corrsum(atria, pointset, query indices, range, exclude) • [c, d] = corrsum(atria, pointset, query indices, range, exclude, bins) Input arguments: • atria - output of nn prepare for pointset (optional) (cf. Section 6.13) • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • range - search range, may be given in one of two ways – If only a single value is given, this value is taken as maximal search radius relative to the attractor diameter (0 < relative range < 1). The minimal search radius is determined automatically be searching for the minimal interpoint distance in the data set. – If a vector of length two is given, the values are interpreted as absolut minimal and maximal search radius. • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches • bins - number of distance values at which the correlation sum is evaluated, defaults to 32 Output arguments: • c - vector of correlation sums, length(c) = bins • d - vector of the corresponding distances at which the correlation sums (stored in c) were computed. d is exponentially spaced, length(c) = bins Example: x = chaosys(25000, 0.025, [0.1 -0.1 0.02], 0); % generate data from Lorenz system x = x(5001:end,:); % discard first 5000 samples due to transient % now compute correlation sum up to five percent of attractor diameter [c,d] = corrsum(x, randref(1,20000, 1000), 0.05, 0); loglog(d,c) % and show the result as log-log plot 38 6.8 corrsum2 - Computation of the correlation sum This is an extended version of the correlation sum algorithm. It tries to accelerate the computation of the correlation sum by using a different number of reference points at each length scale. For large length scales, only a few number of reference points will be used since for this scale, quite a lot of neighbors will fall within this range (and also the search time will be high). The smaller the length scale, the more reference points are used. The algorithm tries to keep the number of pairs found within each range roughly constant at Npairs to ensure a good statistic even for the smallest length scales. However, the number of reference points actually used may be limited to be within [Nref min Nref max] to give at least some control to the user. All reference points are chosen randomly from the data set without reoccurences of the same index. Syntax: • [c, d, e, f, g] = corrsum(pointset, Npairs, range, exclude) • [c, d, e, f, g] = corrsum(pointset, Npairs, range, exclude, bins) • [c, d, e, f, g] = corrsum(pointset, Npairs, range, exclude, bins, opt flag) • [c, d, e, f, g] = corrsum(atria, pointset, Npairs, range, exclude) • [c, d, e, f, g] = corrsum(atria, pointset, Npairs, range, exclude, bins) • [c, d, e, f, g] = corrsum(atria, pointset, Npairs, range, exclude, bins, opt flag) Input arguments: • atria - output of nn prepare for pointset (optional) (cf. Section 6.13) • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • Npairs - Number of pairs to find within each length scale. The algorithm will adapt the number of reference points while computing the correlation sum. Reference points are chosen randomly from the pointset. Optionally, a vector of the form [Npairs Nref min Nref max] may be given. For no length scale less than Nref min reference points will be used. Additionally, not more than Nref max reference points will be used at all. • range - search range, may be given in one of two ways – If only a single value is given, this value is taken as maximal search radius relative to attractor diameter (0 < relative range < 1). The minimal search radius is determined automatically be searching for the minimal interpoint distance in the data set. – If a vector of length two is given, the values are interpreted as absolut minimal and maximal search radius. • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. E.g. if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. exclude = 0 means : exclude self-matches • bins - number of distance values at which the correlation sum is evaluated, defaults to 32 • opt flag - optional flag to control the algorithm: – 0 - Use euclidian distance, be verbose, don’t allow to count a pair of points twice – 1 - Use maximum distance, be verbose, don’t allow to count a pair of points twice 39 – 2 - Use euclidian distance, be verbose, allow to count a pair of points twice – 3 - Use maximum distance, be verbose, allow to count a pair of points twice – 4 - Use euclidian distance, be silent, don’t allow to count a pair of points twice – 5 - Use maximum distance, be silent, don’t allow to count a pair of points twice – 6 - Use euclidian distance, be silent, allow to count a pair of points twice – 7 - Use maximum distance, be silent, allow to count a pair of points twice If the preprocessing output atria is given, the type of metric used to create this overrides the settings by opt flag. Output arguments: • c - vector of correlation sums, length(c) = bins • d - vector of the corresponding distances at which the correlation sums (stored in c) where computed. d is exponentially spaced, length(c) = bins • e - vector of the number of pairs found within this range, length(e) = bins • f - vector of the number of total pairs that were tested, length(f) = bins • g - vector containing the indices of the reference points actually used by the algorithm. Example: x = chaosys(25000, 0.025, [0.1 -0.1 0.02], 0); x = x(5001:end,:); % discard first 5000 samples due to transient % now compute correlation sum up to five percent of attractor diameter [c,d] = corrsum2(x,[1000 100 2000], 0.05, 200); loglog(d,c) % and show the result as log-log plot 6.9 fnearneigh - Fast nearest neighbor search fnearneigh is based on the advanced triangle inequality algorithm ATRIA. However, it does not support approximate queries. The functionality of fnearneigh is almost the same as that of nn search (cf. Section 6.14), so fnearneigh might become obsolete in future versions of TSTOOL. Syntax: • [index, distance] = fnearneigh(pointset, query points, k) • [index, distance] = fnearneigh(pointset, query indices, k, exclude) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • query points - a R by D double matrix containing the coordinates of the query points, organized as R points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • k - number of nearest neighbors to be determined 40 • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • index - a matrix of size R by k which contains the indices of the nearest neighbors. Each row of index contains k indices of the nearest neighbors to the corresponding query point. • distance - a matrix of size R by k which contains the distances of the nearest neighbors to the corresponding query points, sorted in increasing order. 6.10 gendimest - Estimate generalized dimension spectrum The Renyi dimension spectrum of a points set can be estimated using information about the distribution of the interpoint distances. Since we are interested in the scaling behaviour of the Renyi information for small distances, we don’t need to compute all interpoint distances, the distances to k nearest neighbors for each reference point are sufficient [150]. Robust estimation is used instead of mean square error fitting. Syntax: • [dimensions, moments] = gendimest(dists, gammas, kmin low, kmin high, kmax) Input arguments: • dists - a matrix of size R by k which contains distances from reference points to their k nearest neighbors, sorted in increasing order. This matrix can be obtained by calling nn search (cf. Section 6.14) or fnearneigh (cf. Section 6.9) on the point set whose dimension spectrum is to be investigated. • gammas - vector of the moment orders • kmin low - first kmin, 1 ≤ kmin low • kmin high - last kmin, kmin low ≤ kmin high < kmax • kmax - highest neigbor order up to which, kmax ≤ k Output arguments: • dimensions - matrix of size length(gammas) by kmin upper-kmin lower+1, holding the dimension estimates • moments (optional) - matrix of size k by length(gammas), storing the computed moments of the neigbor distances Example: x = chaosys(25000, 0.025, [0.1 -0.1 0.02], 0); % generate data from Lorenz system x = x(5001:end,:); % discard first 5000 samples due to transient [nn, dist] = fnearneigh(x, randref(1, 20000, 1000), 128, 0); gammas = -5:0.5:5; gedims = gendimest(dist, gammas, 8, 8, 128); plot(1-gammas./gedims’, gedims) xlabel(’q’);ylabel(’D_q’);title(’Renyi dimension’) 41 6.11 henon - Generate henon time-series Generate time series by iterating the henon map. Syntax: • x = henon(length, [a b xo yo]) Input arguments: • length - number of samples to generate • [a b xo yo] - vector of parameters and initial conditions Output arguments: • x - vector of size D Example: x = henon(500, [-1.4 0.3 0.2 0.12]); plot(x(:,1), x(:,2), ’.’); 6.12 largelyap - Compute separation of nearby trajectories largelyap is an algorithm very similar to the Wolf algorithm [90] , it computes the average exponential growth of the distance of neighboring orbits via the prediction error. The increase of the prediction error vs the prediction time allows an estimation of the largest lyapunov exponent. Syntax: • x = largelyap(pointset, query indices, taumax, k exclude) • x = largelyap(atria, pointset, query indices, taumax, k exclude) Input arguments: • atria - output of nn prepare for pointset (optional) (cf. Section 6.13) • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • taumax - maximal time shift • k - number of nearest neighbors to compute • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • x - vector of length taumax+1, x(tau) = 1/Nref * sum(log2(dist(reference point + tau, nearest neighbor + tau)/dist(reference point, nearest neighbor))) [146] 42 6.13 nn prepare - Do nearest neighbor preprocessing The intention of this mex-file was to reduce the computational overhead of preprocessing for nearest neighbor or range searching. With nn prepare it is possible to do the preprocessing for a given point set only once and save the created tree structure into a Matlab variable. This Matlab variable, usually called atria, can then be used for repeated neighbor searches on the same point set. Most mex-files that rely on nearest neighbor or range search offer the possibility to use this variable atria as optional input argument. However, if the underlying point set is altered in any way, the proprocessing has to be repeated for the new point set. If the preprocessing output does not belong to the given point set, wrong results or program termination may occur. Syntax: • atria = nn prepare(pointset) • atria = nn prepare(pointset, metric) • atria = nn prepare(pointset, metric, clustersize) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • metric - (optional) either ’euclidian’ or ’maximum’ (default is ’euclidian’) • clustersize - (optional) threshold for clustering algorithm, defaults to 64 Example: pointset = rand(40000, 3); atria = nn_prepare(pointset); [c, d] = corrsum(atria, pointset, 1:17:40000, 0.05, 0); plot(log(d), log(c)) D = takens_estimator(atria, pointset, 1:17:40000, 0.05, 0) 6.14 nn search Syntax: • [index, distance] = nn search(pointset, atria, query points, k) • [index, distance] = nn search(pointset, atria, query points, k, epsilon) • [index, distance] = nn search(pointset, atria, query indices, k, exclude) • [index, distance] = nn search(pointset, atria, query indices, k, exclude, epsilon) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • atria - output of (cf. Section 6.13) nn prepare for pointset 43 • query points - a R by D double matrix containing the coordinates of the query points, organized as R points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • k - number of nearest neighbors to be determined • epsilon - (optional) relative error for approximate nearest neighbors queries, defaults to 0 (= exact search) • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • index - a matrix of size R by k which contains the indices of the nearest neighbors. Each row of index contains k indices of the nearest neighbors to the corresponding query point. • distance - a matrix of size R by k which contains the distances of the nearest neighbors to the corresponding query points, sorted in increasing order. 6.15 predict State space based prediction using nearest neighbors. The algorithms computes one or more nearest neighbors to an initial state vector. The images of the nearest neighbors are used to estimate to image of the initial state vector. The next iteration uses the previously computed image as new initial state vector [145]. Syntax: • x = predict(pointset, length, k, stepsize, mode) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • length - number of iterations (length of prediction) • k - number of nearest neighbors • stepsize - prediction stepsize, usually one • mode - (optional) method to estimate image of initial state vector – 0 - direct prediction, no weight is applied to neighbors – 1 - direct prediction, biquadratic weight is applied to neighbors – 2 - integrated prediction, no weight is applied to neighbors – 3 - integrated prediction, biquadratic weight is applied to neighbors Output arguments: • x - data set as double matrix, size length by D 44 6.16 range search Syntax: • [count, neighbors] = range search(pointset, atria, query points, r) • [count, neighbors] = range search(pointset, atria, query indices, r, exclude) Input arguments: • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • atria - output of (cf. Section 6.13)nn prepare for pointset • query points - a R by D double matrix containing the coordinates of the query points, organized as R points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points • r - range or search radius (r > 0) • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • count - a vector of length R contains the number of points within distance r to the corresponding query point • neighbors - a Matlab cell structure of size R by 2 which contains vectors of indices and vectors of distances to the neighbors for each given query point. This output argument can not be stored in a standard Matlab matrix because the number of neighbors within distance r is not the same for all query points. The vectors if indices and distances for one query point have exactly the length that is given in count. The values in the distances vectors are not sorted.. 6.17 return time return time may be used to find hidden periodicity in multivariate data, e.g. embedded time series data. It computes a histogram of return times. For any given reference point, return time calculates the time span until the time series returns to that location in phase space (by means of nearest neighbors). A histogram of these time spans is computed. Strong peaks in this histogram might be a sign of periodicity in the data. Syntax: • r = return time(pointset, query indices, k, max time, exclude) • r = return time(atria, pointset, query indices, k, max time, exclude) Input arguments: 45 • atria - output of nn prepare for pointset (optional) (cf. Section 6.13) • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • k - number of nearest neighbors to be determined • max time - integer scalar, gives an upper limit for return times that should be considered. • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For example if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • r - vector of length max time, containing the histogram of return times 6.18 takens estimator Syntax: • D = takens estimator(pointset, query indices, relative range, exclude) • D = takens estimator(atria, pointset, query indices, relative range, exclude) Input arguments: • atria - output of nn prepare for pointset (optional) (cf. Section 6.13) • pointset - a N by D double matrix containing the coordinates of the point set, organized as N points of dimension D • query indices - query points are taken out of the pointset, query indices is a vector of length R which contains the indices of the query points (indices may vary from 1 to N) • relative range - search radius, relative to attractor diameter (0 < relative range < 1) • exclude - in case the query points are taken out of the pointset, exclude specifies a range of indices which are omitted from search. For examples if the index of the query point is 124 and exclude is set to 3, points with indices 121 to 127 are omitted from search. Using exclude = 0 means: exclude self-matches Output arguments: • D - scalar value, estimation of correlation dimension 46 6.19 tentmap - Generate tentmap time-series Generate samples of the generalized iterated tentmap. Syntax: • x = tentmap(length, [h e s x0]) Input arguments: • length - number of samples to generate • [h e s x0] - vector of parameters and initial conditions Output arguments: • x - time series Example: x = tentmap(500, [0 1 0.97 rand(1,1)]); plot(x) plot(x(1:end-1), x(2:end), ’.’) 47 6.20 Class signal 6.20.1 Overview Class signal is TSTOOL’s main class. Objects of this type model real world signals. A signal does not only store the pure sample values, it holds much more information like axes, units of sample values or the axes units, and even more descriptive information like labels, command lines and a processing history. The majority of functions in the tstoolbox take a signal as input argument and return a processed signal as output. This allows for combining or chaining of several processing steps in order to get the desired output. 6.20.2 Attributes • xaxes cellarray of at least one object of type achse • core object of type core (cf. Section 6.22) • description object of type description (cf. Section 6.21) 6.20.3 Member functions 6.20.3.1 abs Syntax: • abs(s) Take absolut value of all data values of signal s. If sample values are complex, abs(s) returns the complex modulus (magnitude) of each sample. 6.20.3.2 acf Syntax: • acf(s, len) Input arguments: • len -length of the fft (optional) Autocorrelation function for real scalar signals, using fft (of length len). If len is ommited a default value is calculated. The maximum of the calculated length is 128. 6.20.3.3 acp Syntax: • acp(s, tau, past, maxdelay, maxdim, nref) Input arguments: 49 • tau - proper delay time for s • past - number of samples to exclude before and after reference index (to avoid correlation effects) • maxdelay - maximal delay (should be much smaller than the lenght of s) (optional) • maxdim - maximal dimension to use (optional) • nref - number of reference points (optional) Auto crossprediction function for real scalar signals for increasing dimension. The default value for maxdelay is 25% of the input signal’s length. The default for maxdim is 8 and for nref it is 10% of the input signal’s length. 6.20.3.4 amutual Syntax: • amutual(s, maxtau, bins) Input arguments: • maxtau - maximal delay (should be much smaller than the lenght of s) (optional) • bins - number of bins used for histogram calculation (optional) Auto mutual information function for real scalar signals, can be used to determine a proper delay time for time-delay reconstruction. The default value for maxtau is 25% of the input signal’s length. The default number of bins is 128. X P (A, B) I= P (A, B) log2 P (A)P (B) 6.20.3.5 amutual2 Syntax: • amutual2(s, len) Input arguments: • len - maximal lag Auto mutual information (average) function for real scalar signals using 128 equidistant partitions. 6.20.3.6 analyze Syntax: • analyze(s, maxdim) Input arguments: • maxdim - analyze will not use a dimension higher than this limit Try to do a automatic analysis procedure of a time series. The time series is embedded using the first zero of the auto mutual information function for the delay time. 50 6.20.3.7 arch Syntax: • [rs, archetypes]=arch(s, na, mode=’normalized’) Input arguments: • na - number of generated archetypes • mode - mode can be one of the following : ’normalized’ , ’mean’, ’raw’ (optional) Archetypal analysis of column orientated data set: • each row of data is one ’observation’, e.g. the sample values of all channels in a multichannel measurement at one point in time • in mode ’normalized’ each column of data is centered by removing its mean and then normalized by dividing through its standard deviation before the covariance matrix is calculated • in mode ’mean’ only the mean of every column of data is removed • in mode ’raw’ no preprocessing is applied to data Default value for mode is ’normalized’. 6.20.3.8 boxdim Syntax: • rs = boxdim(s, bins) Input arguments: • s - data points (row vectors) • bins - maximal number of partition per axis (optional) Compute the boxcounting (capacity) dimension of a time-delay reconstructed timeseries s for dimensions from 1 to D, where D is the dimension of the input vectors using boxcounting approach. The default number of bins is 100. 6.20.3.9 cao Syntax: • [E1, E2] = cao(s, maxdim, tau, NNR, Nref) Input arguments: • s - scalar input signal • maxdim - maximal dimension • tau - delay time • NNR - number of nearest neighbor to use • Nref - number of reference points (-1 means: use all points) Estimate minimum embedding dimension using Cao’s method. The second output argument, E2, can be used to distinguish between deterministic and random data. 51 6.20.3.10 center Syntax: • center(s) Center signal by removing it’s mean. 6.20.3.11 corrdim Syntax: • rs = corrdim(s, bins) Input arguments: • s - data points (row vectors) • bins - maximal number of partition per axis (optional) Compute the correlation dimension of a time-delay reconstructed timeseries s for dimensions from 1 to D, where D is the dimension of the input vectors using boxcounting approach. The default number of bins is 100. 6.20.3.12 corrsum Syntax: • rs = corrsum(s, n, range, past, bins) Input arguments: • n - number of randomly chosen reference points (n == -1 means: use all points) • range - maximal relative search radius (relative to attractor size) 0..1 • past - number of samples to exclude before and after each reference index • bins - number of bins (optional) Compute scaling of correlation sum for time-delay reconstructed timeseries s (Grassberger-Proccacia Algorithm), using fast nearest neighbor search. Default number of bins is 20. 6.20.3.13 corrsum2 Syntax: • rs = corrsum2(s, npairs, range, past, bins) Input arguments: • npairs - number of pairs per bins • range - maximal relative search radius (relative to attractor size) 0..1 • past - number of samples to exclude before and after each reference index • bins - number of bins (optional), defaults to 32 Compute scaling of correlation sum for time-delay reconstructed timeseries s (Grassberger-Proccacia Algorithm), using fast nearest neighbor search. 52 6.20.3.14 crosscorrdim Syntax: • rs = crosscorrdim(s, s2, n, range, past, bins) Input arguments: • n - number of randomly chosen reference points (n == -1 means : use all points) • range - maximal relative search radius (relative to size of data set s2) 0..1 • past - number of samples to exclude before and after each reference index • bins - number of bins (optional) Compute scaling of cross-correlation sum for time-delay reconstructed timeseries s against signal s2 (with same dimension as s), using fast nearest neighbor search. Reference points are taken out of signal s, while neigbors are searched in s2. The default number of bins is 32. 6.20.3.15 cut Syntax: • rs = cut(s, dim, start, stop) Input arguments: • dim - dimension along which the signal is cutted • start - position where to start the cut • stop - position where to stop (optional) Cut a part of the signal. If stop is ommited only the data at start is cutted. 6.20.3.16 db Syntax: • db(s, dbmin) Compute decibel values of signal relative to a reference value that is determined by the signal’s yunit values below dbmin are set to dbmin. If dbmin is ommited it is set to -120. 6.20.3.17 delaytime Syntax: • tau = delaytime(s, maxdelay, past) Input arguments: • maxdelay - maximal delay time • past - ? Compute optimal delaytime for a scalar timeseries with method of Parlitz and Wichard. 53 6.20.3.18 diff Syntax: • diff(s, nth) Compute the nth numerical derivative along dimension 1. s has be to sampled equidistantly. 6.20.3.19 dimensions Syntax: • [bc,in,co] = dimensions(s, bins) Input arguments: • s - data points (row vectors) • bins - maximal number of partition per axis, default is 100 Output arguments: • bc - scaling of boxes with partititon sizes (log2 − log2 ) • in - scaling of information with partititon sizes (log2 − log2 ) • co - scaling of correlation with partititon sizes (log2 − log2 ) Compute boxcounting, information and correlation dimension of a time-delay reconstructed timeseries s for dimensions from 1 to D, where D is the dimension of the input vectors using boxcounting approach. Scale data to be within 0 and 1. Give a sortiment of (integer) partitionsizes with almost exponential behaviour. 6.20.3.20 display 6.20.3.21 embed Syntax: • emb = embed(s, dim, delay, shift, windowtype) Input arguments: • dim - embedding dimension • delay - time delay (optional) • shift - shift for two sequent time delay vectors (optional) • windowtype - type of window (optional) Output arguments: • emb - n by dim array, each row contains the coordinates of one point Embeds signal s with embedding dimension dim and delay delay (in samples). s must be a scalar time series. The default values for dim and delay are equal to one. The default value for windowtype is ’Rect’, which is currently the only possible value. 54 6.20.3.22 fft Syntax: • f = fft(s) Output arguments: • f - n by 2 array, the first column contains the magnitudes, the second one the phases. Fourier transform of scalar signal s. 6.20.3.23 filterbank Syntax: • filterbank(s, depth, filterlen) Filter scalar signal s into 2depth bands of equal bandwith, using maximally flat filters. 6.20.3.24 firstmax Syntax: • [xpos, unit] = firstmax(s) Give information about first local maximum of scalar signal s. 6.20.3.25 firstmin Syntax: • [xpos, unit] = firstmin(s) Give information about first local minimum of scalar signal s. 6.20.3.26 firstzero Syntax: • [xpos, unit] = firstzero(s) Give information about first zero of scalar signal s, using linear interpolation. 55 6.20.3.27 fracdims Syntax: • rs = fracdims(s, kmin, kmax, Nref, gstart, gend, past, steps) • rs = fracdims(s, kmin, kmax, Nref, gstart, gend, past) • rs = fracdims(s, kmin, kmax, Nref, gstart, gend) Input arguments: • kmin - minimal number of neighbors for each reference point • kmax - maximal number of neighbors for each reference point • Nref - number of randomly chosen reference points (n == -1 means : use all points) • gstart - starting value for moments • gend - end value for moments • past - (optional) number of samples to exclude before and after each reference index, default is 0 • steps - (optional) number of moments to calculate, default is 32 Compute fractal dimension spectrum D(q) using moments of neighbor distances for time-delay reconstructed timeseries s. Do the main job - computing nearest neighbors for reference points. 6.20.3.28 getaxis Syntax: • a = getaxis(s, dim) Get one of the currend xaxes. 6.20.3.29 gmi Syntax: • gmi(s, D, eps, NNR, len, Nref) Input arguments: • D• eps • NNR • len • Nref Generalized mutual information function for a scalar time series 56 6.20.3.30 histo Syntax: • histo(s, partitions) Histogram function using equidistantly spaced partitions. 6.20.3.31 infodim Syntax: • rs = infodim(s, bins) Input arguments: • s - data points (row vectors) • bins - maximal number of partition per axis, default is 100 Compute the information dimension of a time-delay reconstructed timeseries s for dimensions from 1 to D, where D is the dimension of the input vectors. Using boxcounting approach. Scale data to be within 0 and 1. Give a sortiment of (integer) partitionsizes with almost exponential behaviour. 6.20.3.32 infodim2 Syntax: • rs = infodim2(s, n, kmax, past) Input arguments: • n - number of randomly chosen reference points (n == -1 means : use all points) • kmax - maximal number of neighbors for each reference point • past - number of samples to exclude before and after each reference index Compute scaling of moments of the nearest neighbor distances for time-delay reconstructed timeseries s. This can be used to calculate information dimension D1. Numerically compute first derivative of log γ(k) after k. 6.20.3.33 int Syntax: • int(s) Numerical integration along dimension 1 signal s has to be sampled equidistantly. 57 6.20.3.34 intspikeint Syntax: • rs = intspikeint(s) Compute the interspike intervalls for a spiked scalar timeseries, using transformation on ranked values. 6.20.3.35 intspikint Syntax: • rs = intspikeint(s) Compute the interspike intervalls for a spiked scalar timeseries, using transformation on ranked values. 6.20.3.36 largelyap Syntax: • rs = largelyap(s, n, stepsahead, past, nnr) Input arguments: • n - number of randomly chosen reference points (-1 means: use all points) • stepsahead - maximal length of prediction in samples • past - exclude • nnr - number of nearest neighbours (optional) Output arguments: • rs Compute the largest lyapunov exponent of a time-delay reconstructed timeseries s, using formula (1.5. of Nonlinear Time-Series Analysis, Ulrich Parlitz 1998 [146]). 6.20.3.37 level adaption Syntax: • level adaption(s, timeconstants, dynamic limit, threshold) Each channel of signal s is independently divided by a scaling factor that adapts to the current level of the samples in this channel. The adaption process is simulated using a cascade of feedback loops (P¨ uschel 1998) which consists of low pass filters with time constants given as second argument to this function. The number of time constants given determines the number of feedback loops that are used. Higher values for time constants will result in slower adaption speed. Short time changes in the signal will be transmitted almost linearily. In each feedback loop, a nonlinear compressing characteristic (see Stefan M¨ unkner 1993) limits the signal values to be within [-dynamic limit dynamic limit]. A low value for dynamic limit will introduce nonlinear distortions to the signal. To prevent the feedback loops from adapting to a zero level (in case all input values are zero), a tiny threshold is given as 4th argument. The scaling factors will not shrink below this threshold. 58 6.20.3.38 localdensity Syntax: • rs = localdensity(s, n, past) Input arguments: • n - number of nearest neighbour to compute • past - a nearest neighbour is only valid if it is as least past timesteps away from the reference point past = 1 means: use all points but ref point itself Uses accelerated searching, distances are calculated with euclidian norm. 6.20.3.39 max Syntax: • [maximum, yunit, xpos, xunit] = max(s) Give information about maximum of scalar signal s. Example: disp(’maximum of signal : ’) disp([’y = ’ num2str(m) ’ ’ label(yunit(s))]); disp([’x = ’ num2str(xpos) ’ ’ label(a)]); 6.20.3.40 medianfilt Syntax: • rs = medianfilt(s, len) Moving median filter of width len samples for a scalar time series (len should be odd). 6.20.3.41 merge Syntax: • merge(signal1, signal2, dB) • merge(signal1, signal2) Input arguments: • signal1, signal2 - Signals • dB - energy ratio, (optional, default = 0) Merges signal s1 and s2 into a new signal with energy ration dB (in decibel) a positive value of dB increases the amount of signal1 in the resulting signal. 59 6.20.3.42 min Syntax: • [minimum, yunit, xpos, xunit] = min(s) Give information about minimum of scalar signal s. Example: disp(’minimum of signal : ’) disp([’y = ’ num2str(m) ’ ’ label(yunit(s))]); disp([’x = ’ num2str(xpos) ’ ’ label(a)]); 6.20.3.43 minus Syntax: • rs=minus(s, offset) • rs=minus(s1,s2) Input arguments: • s, s1, s2 - signal object • offset - scalar value Calculate difference of signals s1 and s2 or substract a scalar value from s. 6.20.3.44 movav Syntax: • rs = movav(s, len, windowtype) • rs = movav(s, len) Moving average of width len (samples) along first dimension. 6.20.3.45 multires Syntax: • rs = multires(s) => scale=3 • rs = multires(s, scale) Multires perform multiresolution analysis. Y = MULTIRES (X,H,RH,G,RG,SC) obtains the SC successive details and the low frequency approximation of signal in X from a multiresolution scheme. The analysis lowpass filter H, synthesis lowpass filter RH, analysis highpass filter G and synthesis highpass filter RG are used to implement the scheme. Results are given in a scale+1 channels. The first scale channels are the details corresponding to the scales 21 to 2scale the last row contains the approximation at scale 2SC . The original signal can be restored by summing all the channels of the resulting signal. 60 6.20.3.46 nearneigh Syntax: • rs = nearneigh(s, n) => past=1 • rs = nearneigh(s, n, past) Input arguments: • n - number of nearest neighbour to compute • past - a nearest neighbour is only valid if it is as least past timesteps away from the reference point. past = 1 means: use all points but ref point itself n nearest neighbour algorithm. Find n nearest neighbours (in order of increasing distances) to each point in signal s uses accelerated searching, distances are calculated with euclidian norm. 6.20.3.47 norm1 Syntax: • rs=norm1(s) => low=0 , upp=1 • rs=norm1(s, low) => upp=1 • rs=norm1(s, low, upp) Scale and move signal values to be within [low,upp]. 6.20.3.48 norm2 Syntax: • rs=norm2(s) Normalize signal by removing it’s mean and dividing by the standard deviation. 6.20.3.49 pca Syntax: • [rs, eigvals, eigvecs] = pca(s) => mode=’normalized’ , maxpercent = 95 • [rs, eigvals, eigvecs] = pca(s, mode) => maxpercent = 95 • [rs, eigvals, eigvecs] = pca(s, mode, maxpercent) Input arguments: • each row of data is one ’observation’, e.g. the sample values of all channels in a multichannel measurement at one point in time • mode can be one of the following : ’normalized’ (default), ’mean’, ’raw’ 61 – in mode ’normalized’ each column of data is centered by removing its mean and then normalized by dividing through its standard deviation before the covariance matrix is calculated – in mode ’mean’ only the mean of every column of data is removed – in mode ’raw’ no preprocessing is applied to data • maxpercent gives the limit of the accumulated percentage of the resulting eigenvalues, default is 95 % Principal component analysis of column orientated data set. 6.20.3.50 plosivity Syntax: • rs = plosivity(s, blen) => flen=1 , thresh=0, windowtype = ’Rect’ • rs = plosivity(s, blen, flen) => thresh=0, windowtype = ’Rect’ • rs = plosivity(s, blen, flen, thresh) => windowtype = ’Rect’ • rs = plosivity(s, blen, flen, thresh, windowtype) Compute plosivity of a spectrogram. See also: window for list of possible window types. 6.20.3.51 plus Syntax: • rs=plus(s, offset) • rs=plus(s1, s2) Add two signals s1 and s2 or add a scalar value offset to s. 6.20.3.52 poincare Syntax: • rs=poincare(s, ref) Compute Poincare-section of an embedded time series the result is a set of vector points with dimension DIM-1, when the input data set of vectors had dimension DIM. The projection is done orthogonal to the tangential vector at the vector with index. 6.20.3.53 power Syntax: • power(s) Calculate squared magnitude of each sample. 62 6.20.3.54 predict Syntax: • rs = predict(s, dim, delay, len) => nnr=1 • rs = predict(s, dim, delay, len, nnr) => mode=0 • rs = predict(s, dim, delay, len, nnr, mode) Input arguments: • dim - dimension for time-delay reconstruction • delay - delay time (in samples) for time-delay reconstruction • len - length of prediction (number of output values) • nnr - number of nearest neighbors to use (default is one) • step - stepsize (in samples) (default is one) • mode: – 0 = Output vectors are the mean of the images of the nearest neighbors – 1 = Output vectors are the distance weighted mean of the images of the nearest neighbors – 2 = Output vectors are calculated based on the local flow using the mean of the images of the neighbors – 3 = Output vectors are calculated based on the local flow using the weighted mean of the images of the neighbors Local constant iterative prediction for scalar data, using fast nearest neighbor search. Four methods of computing the prediction output are possible. 6.20.3.55 predict2 Syntax: • rs = predict2(s, len, nnr, step, mode) Input arguments: • len - length of prediction (number of output values) • nnr - number of nearest neighbors to use (default is one) • step - stepsize (in samples) (default is one) • mode: – 0 = Output vectors are the mean of the images of the nearest neighbors – 1 = Output vectors are the distance weighted mean of the images of the nearest neighbors – 2 = Output vectors are calculated based on the local flow using the mean of the images of the neighbors – 3 = Output vectors are calculated based on the local flow using the weighted mean of the images of the neighbors Local constant iterative prediction for phase space data (e.g. data stemming from a time delay reconstruction of a scalar time series), using fast nearest neighbor search. Four methods of computing the prediction output are possible. 63 6.20.3.56 rang Syntax: • rs = rang(s) Transform scalar time series to rang values. 6.20.3.57 removeaxis Syntax: • s = removeaxis(s, dim) Remove axis one of the current xaxes. No bound checking for dim. 6.20.3.58 return time Syntax: • rs = return time(s, nnr, maxT) => past=1 • rs = return time(s, nnr, maxT, past) • rs = return time(s, nnr, maxT, past, N) Input arguments: • nnr - number of nearest neighbors • maxT - maximal return time to consider • past - a nearest neighbor is only valid if it is as least past timesteps away from the reference point past = 1 means: use all points but tt ref point itself • N - number of reference indices Compute histogram of return times. 6.20.3.59 reverse Syntax: • rs=reverse(s) Reverse signal along dimension 1. 6.20.3.60 rms Syntax: • rs = rms(s) Calculate root mean square value for signal along dimension 1. 64 6.20.3.61 scale Syntax: • scale(signal, factor) Scale signal by factor f. 6.20.3.62 scalogram Syntax: • rs = scalogram(s) => scalemin=0.1 • rs = scalogram(s, scalemin) => scalemax=1 • rs = scalogram(s, scalemin, scalemax) => scalestep=0.1 • rs = scalogram(s, scalemin, scalemax, scalestep) => mlen=10 • rs = scalogram(s, scalemin, scalemax, scalestep, mlen) Scalogram of signal s using morlet wavelet. See also: spec2. 6.20.3.63 setaxis Syntax: • s = setaxis(s, dim, achse) Change one of the current xaxes. 6.20.3.64 setunit Syntax: • s = setunir(s, dim, u) Change unit of one of the current xaxes. 6.20.3.65 shift Syntax: • s = shift(s, distance) (dim=1) • s = shift(s, distance, dim) shift signal on axis No. dim by distance (measured in the unit of the axis) to the right 65 6.20.3.66 signal Syntax: • s = signal(array) creates a new signal object from a data array array the data inside the object can be retrieved with x = data(s); • s = signal(array, achse1, achse2, ...) creates a new signal object from a data array array, using achse1 etc. as xachse entries • s = signal(array, unit1, unit2, ...) creates a new signal object from a data array ’array’, using unit1 etc. to create xachse objects • s = signal(array, samplerate1, samplerate2, ...) creates a new signal object from a data array array, using as xunit ’s’ (second) and scalar samplerate1 as samplerate(s) A signal object contains signal data, that is a collection of real or complex valued samples. A signal can be one or multi-dimensional. The number of dimensions is the number of axes that are needed to describe the the data. An example for an one-dimensional signal is a one-channel measurement (timeseries), or the power spectrum of a one-channel measurement. An example for a two-dimensional signal is a twelve-channel measurement, with one time axis and a ’channel’ axis. Another example for a two-dimensional signal is a short time spectrogramm of a time series, where we have a time axis and a frequency axis. Each axis can have a physical unit(e.g. ’s’ or ’Hz’), a starting point and a step value. E.g. if a time-series is sampled with 1000 Hz, beginning at 1 min 12 sec, the unit is ’s’, the starting point is 72 and the step value (delta) is 0.001. But not only the axes have physical units, also the sample value themselve can have a unit, maybe ’V’ or ’Pa’, depending on what the sampled data represent (=¿ yunit) All units are stored as objects of class ’unit’, all axes are stored as objects of class ’achse’ (this somewhat peculiar name was chosen because of conflicts with reserved matlab keywords ’axis’ and ’axes’, which otherwise would have been the first choice). Example for creating a 2-dimensional signal with y-unit set to ’Volt’, the first dimension’s unit is ’second’ (time), the second dimension’s unit is ’n’ (Channels). Examples: • tmp = rand(100, 10); s = signal(tmp, unit(’s’), unit(’n’)); s = setyunit(s, unit(’V’)); s = addcomment(s, ’Example signal with two dimensions’) • Loading from disk s = signal(filename) loads a previously stored signal object • Importing from other file formats: ASCII: s = signal(’data/spalte1.dat’, ’ASCII’) WAVE: s = signal(’data/Sounds/hat.wav’, ’WAVE’) AU (SUN AUDIO): s = signal(’data/Sounds/hat.au’, ’AU’) (old) NLD-Format : s = signal(’test.nld’, ’NLD’) 66 6.20.3.67 spacing • v = spacing(s) (dim=1) • v = spacing(s, dim) return spacing values for xaxis nr. dim 6.20.3.68 spec Syntax: • rs = spec(s) compute power spectrum for real valued scalar signals. Multivariate signals are accepted but may produce unwanted results as only the spectrum of the first column is returned. 6.20.3.69 spec2 Syntax: • rs = spec2(s) Input Arguments: • fensterlen - size of window (optional) • fenster - window type (optional) • vorschub - shift in samples (optional) spectrogramm of signal s using short time fft Examples: view(spec2(sine(10000, 1000, 8000), 512, ’Hanning’)) 6.20.3.70 stts Syntax: • rs = stts(s, I) (J=0, K=1, L=1) • rs = stts(s, I, J) (K=1, L=1) • rs = stts(s, I, J, K) (L=1) • rs = stts(s, I, J, K, L) Input Arguments: • s - input data set of N snapshots of length M, given as N by M matrix • I - number of spatial neighbours • J - number of temporal neighbours (in the past) • K - spatial shift (= spatial delay) • L - temporal delay Spatiotemporal prediction conforming to U. Parlitz, NONLINEAR TIME-SERIES ANALYSIS Chapter 1.10.2.1. 67 6.20.3.71 sttserror Syntax: • rs = sttserror(s1, s2) Input Arguments: • s1 - original signal • s2 - predicted signal compute error function for prediction of spatial-temporal systems see U. Parlitz "Nonlinear Time Series Analysis", Section 1.10.2.2 Eq. 1.10 6.20.3.72 surrogate1 Syntax: • rs = surrogate1(s) create surrogate data for a scalar time series by randomizing phases of fourier spectrum see : James Theiler et al.’Using Surrogate Data to Detect Nonlinearity in Time Series’, APPENDIX : ALGORITHM I 6.20.3.73 surrogate2 Syntax: • rs = surrogate2(s) create surrogate data for a scalar time series see : James Theiler et al.’Using Surrogate Data to Detect Nonlinearity in Time Series’, APPENDIX : ALGORITHM II 6.20.3.74 surrogate3 Syntax: • rs = surrogate3(s) create surrogate data for a scalar time series by permuting samples randomly 6.20.3.75 surrogate test Syntax: • rs=surrogate test(s, ntests, method,func) Input Arguments: • s - has to be a real, scalar signal 68 • ntests - is the number of surrogate data sets to create • method - method to generate surrogate data sets: – 1: surrogate1 – 2: surrogate2 – 3: surrogate3 • func - string with matlab-code, have to return a signal object with a scalar time series. The data to process is a signal object referred by the qualifier s (see example). Output Arguments: • rs is a signal object with a three dimensional time series. The first component is the result of the func function applied to the original data set s. The second component is the mean of the result of the func function applied to the ntests surrogate data sets. The third component is the standard deviation. There is a special plothint (’surrerrorbar’) for the view function to show this result in the common way. surrogate test runs an automatic surrogate data test task. It generates ntests surrogate data sets an performs the func function to each set. func is a string with matlab-code who returns a signal s with a scalar time series. Example: st = surrogate_test(s, 10, 1, 1, ’largelyap(embed(s,3,1,1), 128,20,10);’); 6.20.3.76 swap Syntax: • rs = swap(s) (exchange dimension 1 and dimension 2) • rs = swap(s, dim1, dim2) Exchange signal’s dimensions (and axes) 6.20.3.77 takens estimator Syntax: • D2 = takens estimator2(s, n, range, past) Input Arguments: • n - number of randomly chosen reference points (n == -1 means : use all points) • range - maximal relative search radius (relative to attractor size) 0..1 • past - number of samples to exclude before and after each reference index Takens estimator for correlation dimension 69 6.20.3.78 tc3 Syntax: • rs = tc3(s,tau,n,method) Input Arguments: • tau - see explaination below • n - number of surrogate data sets to generate • method - method to generate the surrogate data sets: – 1: surrogate1 – 2: surrogate2 – 3: surrogate3 Output Arguments: • rs is a row vector, returned as signal object. The first item is the TC3 value for the original data set s. The following n values are the TC3 values for the generated surrogates. There exist a special plothint (’surrbar’) for the view function to show this kind of result in the common way. This function calculates a special value for the original data set and the n generated surrogate data sets. The TC3 value is defined as followed: TC3 ({xn }, τ ) = hxn xn−τ xn−2τ i 3 |hxn xn−τ i| 2 In terms of surrogate data test this is a test statistics for higher order moments. The original tc3 function is located under utils/tc3.m and use simple matlab vectors. 6.20.3.79 trend Syntax: • rs = trend(s, len) trend correction calculate moving average of width len (samples) for a scalar time series (len should be odd) and remove the result from the input signal 6.20.3.80 trev Syntax: • rs = trev(s,tau,n,method) Input Arguments: • tau - see explaination below • n - number of surrogate data sets to generate 70 • method - method to generate the surrogate data sets: – 1: surrogate1 – 2: surrogate2 – 3: surrogate3 Output Arguments: • rs is a row vector, returned as signal object. The first item is the TREV value for the original data set s. The following n values are the TREV values for the generated surrogates. There exist a special plothint (’surrbar’) for the view function to show this kind of result in the common way. This function calculates a special value for the original data set and the n generated surrogate data sets. The TREV value is defined as followed: TREV ({xn }, τ ) = h(xn − xn−τ )3 i 3 h(xn − xn−τ )2 i 2 In terms of surrogate data test this is a test statistics for time reversibility. The original trev function is located under utils/trev.m and use simple matlab vectors. 6.20.3.81 upsample Syntax: • rs = upsample(s, factor, method) Input Arguments: • method may be one of the following : – ’fft’ – ’spline’ – ’akima’ – ’nearest’ – ’linear’ – ’cubic’ • s has be to sampled equidistantly for fft interpolation Change sample rate of signal s by one-dimensional interpolation 6.20.3.82 view Syntax: • view(signal) (fontsize=12) • view(signal, fontsize) • view(signal, fontsize, figurehandle) 71 Signal viewer that decides from the signal’s attributes which kind of plot to produce, using the signal’s plothint entry to get a hint which kind of plot to produce Possible plothints are: • ’graph’ • ’bar’ • ’surrbar’ • ’surrerrorbar’ • ’points’ • ’xyplot’ • ’xypoints’ • ’scatter’ • ’3dcurve’ • ’3dpoints’ • ’spectrogram’ • ’image’ • ’multigraph’ • ’multipoints’ • ’subplotgraph’ 6.20.3.83 write Syntax: • write(s, filename) (writes in TSTOOL’s own file format) • write(s, filename, ’ASCII’) • write(s, filename, ’WAV’) (RIFF WAVE FORMAT) • write(s, filename, ’AU’) (SUN AUDIO FORMAT) • write(s, filename, ’NLD’) (old NLD FORMAT) • write(s, filename, ’SIPP’) (si++ file format) writes a signal object to file filename (uses matlab’s file format) 6.21 Class description 6.21.1 Overview Class description is the second base class of class signal (cf. Section 6.20). An object of type description stores all descriptive information belonging to a signal. 72 6.21.2 Attributes • label - string • name - string • type - string • plothint - string • comment - object of type list (cf. Section 6.25) • history - object of type list (cf. Section 6.25) • creator - string • yname - string • yunit - object of type unit (cf. Section 6.24) • commandlines - object of type list (cf. Section 6.25) • optparam - cell array, may be used to store optional information 6.21.3 Member functions 6.21.3.1 addcommandlines adds new commandline to list of commands that have been applied to that signal example 1 addcommandlines(s, ’s = spec2(s’, 512, ’Hanning’ )) will add ’s = spec2(s, 512, ’Hanning’);’ to the list of applied commands example 2 len = 512; text = ’Hanning’; addcommandlines(s, ’s = spec2(s’, 512, ’Hanning’ )) will add ’s = spec2(s, 512, ’Hanning’);’ to the list of applied commands 6.21.3.2 addcomment adds new comment to current list of comments 6.21.3.3 addhistory adds text to current history list always the current time and date is written into the first line 6.21.3.4 commandlines 6.21.3.5 comment 6.21.3.6 creator 6.21.3.7 description description class constructor Syntax: • d = description() • d = description(name) 73 • d = description(name, type) • d = description(yunit) An object of class description contains auxiliary descriptive information for a signal, e.g. information about data unit, creator, how the signal should be plotted, a user specified comment text, a processing history and the commandlines that were used to generate this signal 6.21.3.8 display description/display 6.21.3.9 history description/history 6.21.3.10 label 6.21.3.11 makescript Syntax: • makescript (signal, scriptfilename) creates a Matlab m-file that contains exactly the the processing steps that have been applied to get the input signal. This gives a kind of macro facility for tstool. Example signal s was calculated through several processing steps from signal s0 (the raw or original signal) Now makescript(s, ’foo.m’) will create a Matlab m-file named foo.m which, applied to s0, will give s. 6.21.3.12 merge Syntax: • d = merge(d1, d2) merge two descriptions Most items are taken from first description. History is taken from both descriptions. This function may be useful when writing binary operators for class signal 6.21.3.13 name description/name Syntax: • n = name(d) Get signal’s name 74 6.21.3.14 newcomment Syntax: • d = newcomment(d, string) • d = newcomment(d, list) Replace old comment with new comment 6.21.3.15 optparams Syntax: • param = optparams(d, nr) get optional parameter number nr 6.21.3.16 plothint 6.21.3.17 setlabel Syntax: • d = setlabel(d, label) the label field of a description is used to give a signal some ’tag’ which remains constant through various processing steps e.g. which topic this signal belongs to 6.21.3.18 setname Syntax: • d = setname(d, name) the name field of a descriptiom is used when the signal is loaded from file, it will not be continued through several processing steps 6.21.3.19 setoptparams Syntax: • d = setoptparams(d, nr, param) set optional parameter number nr 6.21.3.20 setplothint 6.21.3.21 settype Syntax: • d = settype(d, string) Set a new type for signal 75 6.21.3.22 setyname Syntax: • d = setyunit(d, string) Set signal’s y-name e.g. d = setyunit(d, ’V’) 6.21.3.23 setyunit Syntax: • d = setyunit(d, unit) • d = setyunit(d, string) Set signal’s y-unit e.g. d = setyunit(d, ’V’) 6.21.3.24 type return signal type (e.g. ’Correlation function’, ’Spectrogram’ etc.) 6.21.3.25 yname return name of the measured data (e.g. ’Heartbeat rate’, ’Current’ etc.) 6.21.3.26 yunit return y-unit of the sampled data values (e.g. Volt, Pa etc.) 6.22 Class core 6.22.1 Overview Class core is a base class of class signal (cf. Section 6.20). An object of type core stores the pure sample values of a signal, without any additional descriptive information. The separation of the numerical and the descriptive part of a signal simplifies the writing of m-files that work on signals. 6.22.2 Attributes • data - double matrix (one, two or multidimensional) TSTOOL stores a one-dimensional time-series always as a row vector ! Rows correspond to the first xaxis, columns to the second ... • dlens - double vector, storing size of data 76 6.22.3 Member functions 6.22.3.1 acf Syntax: • acf(cin, m) Input Arguments: • cin core object • m fft-length acf calculates the autocorrelation function of cin via fft of length m. 6.22.3.2 amutual2 Syntax: • amutual(cin, len) Input Arguments: • cin core object • len maximal lag amutual2 calculates the mutual information of a time series against itself, with increasing lag uses equidistant partitioning to compute histograms. 6.22.3.3 compare Syntax: • compare(c1,c2, tolerance) Input Arguments: • c1,c2 core object of two signals • tolerance tolerance of the signals’s RMS value (default tolerance=1e-6) compare compare two signals whether they have equal values slight differences due to rounding errors are ignored depending on the value of tolerance when signals are found to be not equal, a zero is returned. 6.22.3.4 core core class constructor Syntax: • c = core(arg) Input Arguments: • arg double array A core object contains the pure data part of a signal object. Methods: ndim dlens data 77 6.22.3.5 data Syntax: • d = data(c, varargin) • c=core object Input Arguments: • varargin - selector string for data-elements in matlab notation Return signal’s data values With no extra arguments, data returns the data array of a signal object Another possible call is : data(signal, ’:,:,1:20’) 6.22.3.6 db Syntax: • cout = db(cin, ref, scf, dbmin) Input Arguments: • cin - core object • ref - reference value • scf - scaling factor • dbmin - minimal db-value compute decibel values to reference value ref and scaling factor (10 or 20) scf 6.22.3.7 diff Syntax: • cout = diff(cin, nth, delta) Input Arguments: • cin - core object • nth - number of derivations • delta - time difference between to signal values nth numerical derivative along dimension 1 when data was sampled equidistantly with samplerate = 1/delta 78 6.22.3.8 display Syntax: • display(c) Input Arguments: • c - core object 6.22.3.9 dlens Syntax: • d=dlens(c, nr) Input Arguments: • c - core object returns sizes of dimensions (same as function ’size’ under matlab) 6.22.3.10 embed Syntax: • cout = embed(cin, dim, delay, shift, windowtype) Input Arguments: • cin - core object • dim - embed dimension • delay - delay time in samples for time delay vectors • shift - shift in samples for two sequent time delay vectors • windowtype - type of window Create time delay vectors with dimension dim, delay is measured in samples The input must be a scalar time series The result is a n by dim array, each row contains the coordinates of one point 6.22.3.11 filterbank Syntax: • filterbank(cin,H,G,ORDER,BASIS) Input Arguments: • H - lowpass filter 79 • G - highpass filter • ORDER - indicates the type of tree: – 0 - band sorting according to the filter bank – 1 - band sorting according to the frequency decomposition • BASIS - desired subband decomposition calculates the Wavelet Packet Transform of cin. It can be obtained using a selection algorithm function. It may be switched from one format to another using CHFORMAT. The different bands are sorted according to ORDER and BASIS. If BASIS is omitted, the output is a matrix with the coefficients obtained from all the wavelet packet basis in the library. Each column in the matrix represents the outputs for a level in the tree. The first column is the original signal. If the length of X is not a power of 2, the columns are zero padded to fit the different lengths. Run the script ’BASIS’ for help on the basis format. See also: IWPK, CHFORMAT, PRUNEADD, PRUNENON, GROWADD, GROWNON. 6.22.3.12 int Syntax: • cout = int(cin, delta) Input Arguments: • cin - core object • delta - time period between two data samples numerical integration along dimension 1 when data was sampled equidistantly with samplerate = 1/delta 6.22.3.13 intermutual Syntax: • intermutual(cin1,cin2,n) Input Arguments: • cin1,cin2 - core objects Calculates the mutual information of cin1 and cin2. 6.22.3.14 isempty Syntax: • r = isempty(s) Input Arguments: • s - core object test if core contains no (valid) data 80 6.22.3.15 medianfilt Syntax: • medianfilt(cin,len) Input Arguments: • cin - core object moving median filter 6.22.3.16 minus Syntax: • minus(c1,c2) Input Arguments: • c1,c2 - core objects subtract c2 from each columns of c1 6.22.3.17 movav Syntax: • movav(cin,len) Input Arguments: • cin - core object • len - average length moving average 6.22.3.18 multires Syntax: • multires(cin,h,rh,g,rg,sc) Input Arguments: • cin - core object 81 6.22.3.19 ndim Syntax: • ndim(c) Input Arguments: • c - core object return number of dimensions, a scalar value has 0 dimensions 6.22.3.20 norm1 Syntax: • cout = norm1(cin,low,upp) Input Arguments: • cin - core object • low - column number • upp - column number normalize each single column of a the core object to be within [low,upp] 6.22.3.21 norm2 Syntax: • cout = norm2(cin) Input Arguments: • cin - core object normalize signal by removing it’s mean and dividing by the standard deviation 6.22.3.22 plus Syntax: • plus(c1,c2) Input Arguments: • c1,c2 - core objects add c2 to each columns of c1 82 6.22.3.23 rang Syntax: • cout = rang(cin) Input Arguments: • cin - core object 6.22.3.24 rms Syntax: • cout = rms(cin) Input Arguments: • cin - core object compute root mean square value of each column of c1 6.22.3.25 scalogram Syntax: • cout = scalogram(cin, smin, smax, sstep, tim) 6.22.3.26 spec Syntax: • cout = spec(cin) Input Arguments: • cin - core object compute power spectrum for real valued signals 6.22.3.27 spec2 Syntax: • cout = spec2(cin, fensterlen, fenster, vorschub) Input Arguments: • cin - core object • fensterlen - window size • fenster - type of window • vorschub - moving step spectrogramm of data using short time fft 83 6.22.3.28 surrogate1 Syntax: • cout = surrogate1(cin) Input Arguments: • cin - core object create surrogate data for a scalar time series by randomizing phases of fourier spectrum see : James Theiler et al.’Using Surrogate Data to Detect Nonlinearity in Time Series’, APPENDIX : ALGORITHM I 6.22.3.29 surrogate2 Syntax: • cout = surrogate2(cin) Input Arguments: • cin - core object create surrogate data for a scalar time series see : James Theiler et al.’Using Surrogate Data to Detect Nonlinearity in Time Series’, APPENDIX : ALGORITHM II 6.22.3.30 surrogate3 Syntax: • cout = surrogate3(cin) Input Arguments: • cin - core object create surrogate data for a scalar time series by permuting samples randomly 6.22.3.31 uminus Syntax: • r = uminus(c) Input Arguments: • c - core object negate time series 84 6.22.3.32 vertcat Syntax: • r = vertcat(c1,c2) Input Arguments: • c1,c2 - core objects catenate two timeseries verticaly 6.23 Class achse 6.23.1 Overview Class achse models an axis, e.g. a time axis or a frequency axis. A signal has a least one axis (if it is a one dimensional signal). A multidimensional signal has several achse objects. An achse object is basically described by an object of class unit and the spacing values. The spacing may be linear, logarithmic or arbitrary (in case of non-uniform sampling). 6.23.1.1 Why is class achse not called class axis ? The names axis and axes are already occupied in Matlab. So, achse, which is the german translation of axis, was used as name for that class. 6.23.2 Attributes • name - string, name of axis (e.g. ’Time’) • quantity - string • unit - object of type unit (cf. Section 6.24) • resolution - string, may be ’linear’, ’logarithmic’ or ’arbitrary’ • first - double value, starting value of this axis • delta - double value, stepwidth for this axis • values - double vector, stores spacing values in case of ’arbitrary’ resolution • opt - cell array, may be used to store optional information 6.23.3 Member functions 6.23.3.1 achse achse class constructor Syntax: • a = achse creates default achse object 85 • a = achse(axs) copies achse object axs into a • a = achse(unt) creates achse object using unit unt, with linear spacing, first = 0, delta = 1 • a = achse(vec) creates achse object with arbitrary spacing, using values in vec as spacing data • a = achse(unt, vec) creates achse object using unit unt with arbitrary spacing, using values in vec as spacing data • a = achse(unt, first, delta) creates achse object with linear spacing, using delta and first • a = achse(unt, first, delta, ’log’) creates achse object with logarithmic spacing, using delta and first achse used to describe the different dimensions (axes) of a signal object. Example: • a = achse(unit(’Hz’), 0.01, 10, ’log’) creates a logarithmic frequency axis with values 0.01 Hz, 0.1 Hz, 1 Hz, 10 Hz • a = achse(label, samplerate) has the same result as a = achse(unit(label), 0, 1/samplerate) see also: delta first horzcat label name quantity resolution samplerate scale setname spacing unit 6.23.3.2 cut Syntax: • a = cut(a, start, stop) Cut a part out of achse a, beginning from index start up to index stop. stop is only needed in case of arbitrary spacing. cut ensures the following: If values = spacing(achse1, N) and N > n then values(n:N) == spacing(cut(achse1, n), N+1-n) See also: horzcat 6.23.3.3 delta 6.23.3.4 display 6.23.3.5 eq Test if achse a and achse b are equal.first is not (!) taken into account for this test. 86 6.23.3.6 first 6.23.3.7 horzcat 6.23.3.8 label 6.23.3.9 name 6.23.3.10 quantity 6.23.3.11 resolution 6.23.3.12 samplerate Syntax: • rate = samplerate(a) samplerate returns samplerate of achse object. 6.23.3.13 scale Syntax: • r = scale(a,f) Scale achses delta by factor f. 6.23.3.14 setdelta Syntax: • a = setdelta(a,f) 6.23.3.15 setfirst Syntax: • a = setfirst(a,f) 6.23.3.16 setname Syntax: • a = setname(a, newname) 6.23.3.17 setunit Syntax: • a = setunit(a,u) 87 6.23.3.18 setvalues Syntax: • a = setvalues(a, v) 6.23.3.19 spacing Syntax: • v = values(a, len) Returns spacing values for linear, logarithmic or arbitary spacing in case of lin. or log. spacing. len values are returned. In case of arbitary spacing, all stored values are returned. 6.23.3.20 unit 6.24 Class unit 6.24.1 Overview Objects of class ’unit’ try to model physical units. It’s is possible to multiply or divide objects of this type. A small database is used to find the right label for compound units. See also : directory @unit/private, file units.mat 6.24.2 Attributes • label - string • name - string • quantity - structure, holding two strings • factor - double value • exponents - vector • dBScale - double value • dBRef - double value • opt - cell array, may be used to store optional information 6.24.3 Member functions 6.24.3.1 char gives the unit’s label (e.g. V for Volt) back. 6.24.3.2 dbref returns reference value for 0 dB when calculating decibel values from data of this unit. 88 6.24.3.3 dbscale returns scaling value when calculating decibel values from data of this unit. dpscale returns either 10 (for power or energy units (e.g. Watt)) or 20 (for all other units (e.g. Volt). 6.24.3.4 display 6.24.3.5 double gives a row vector which’s first element contains the unit’s factor and the remaining elements contain the exponents of the SI basic units. 6.24.3.6 eq 6.24.3.7 exponents returns dimension exponents of unit q. 6.24.3.8 factor returns factor of unit q. 6.24.3.9 6.24.3.10 label mpower Syntax: • mpower(u,p) take unit u to power p, p must be a scalar. 6.24.3.11 mrdivide 6.24.3.12 mtimes 6.24.3.13 name returns name of unit q. 6.24.3.14 quantity returns quantity name of unit q. If argument which is omitted, the english quantity name will be returned. 89 6.24.3.15 unit unit class constructor Class unit tries to modell physical units a physical unit is mainly can be described by the exponents of the basic SI units, namely mass, length, time, current, temperature, luminal intensity, mole and plane angle. Each unit belongs to a quantity, e.g. the unit s (second) is used when measuring the quantity TIME. Each unit has a name, e.g. ’Ampere’, ’Volt’ , ’Joule’, ’hour’, and an abbreviation, called label (’A’, ’V’, ’J’, ’h’). Unfortunately, the correspondence between these items is not always bijectiv to find corresponding items, a table of units in the file units.mat is used. A unit object can be created with different types of arguments: • by giving the label: unit(’Hz’) looks up the remaining data (exponents, name, quantity) in the table • by giving the exponents Some arithmetic can be done with units: • units can be multiplied unit(’V’) * unit(’A’) = unit(’Watt’) • or taken to an integer or rational power unit(’m’)2 6.25 Class list 6.25.1 Overview Simple list of strings, used in class description (cf. Section 6.21). 6.25.2 Attributes • data : cellarray of strings • len : double value, counts number of strings in data 6.25.3 Member functions 6.25.3.1 append Syntax: • list = append(list, string) • list = append(list, list) Add string(s) to existing list. 6.25.3.2 cellstr cellstr return cell array of strings from list l. 90 6.25.3.3 char returns a char array from list l. 6.25.3.4 display 6.25.3.5 get Syntax: • s = get(l, nr) returns string number nr from list l. 6.25.3.6 length Syntax: • len = length(l) returns the number of strings in list l. 6.25.3.7 list Syntax: • l = list creates empty list • l = list(’Hello world’) create list with one entry, ’Hello world’ • l = list(’Hello’, ’My’ , ’World’) create list with three entries • l = list(’Hello’, ’My’ , ’World’) create list with three entries An object of type list contains a list of strings. 6.25.3.8 sort sort list l in increasing order. 91 Chapter 7 Frequently asked questions 7.1 Questions 1. Introduction and general information (cf. Section 7.2.1) • What is TSTOOL ? (cf. Section 7.2.1.1) • What software is required to run TSTOOL ? (cf. Section 7.2.1.2) • On which systems does TSTOOL run ? (cf. Section 7.2.1.3) • What about Octave or other Matlab like programming environments ? (cf. Section 7.2.1.4) 2. Installation of TSTOOL • All lines in the OpenTSTOOL/tstoolbox/mex/*.m are comments, is this right? (cf. Section 7.2.2.1) • Where are the precompiled Mex-Files? (cf. Section 7.2.2.2) • There are more than one file called e.g. amutual.m, why? (cf. Section 7.2.2.3) • What does the error message ”Attempt to execute SCRIPT . . . as a function.” mean? (cf. Section 7.2.2.4) • What does the error message ”This application has failed to start because the application configuration is incorrect.” mean? (cf. Section 7.2.2.5) 3. Working with TSTOOL (cf. Section 7.2.3) • How do I create a signal from my time-series data ? (cf. Section 7.2.3.1) • How do I create a signal with logarithmic spacing ? (cf. Section 7.2.3.2) • How do I create a signal from non-uniformly sampled data ? (cf. Section 7.2.3.3) • How do I change the type of plot that I get with view ? (cf. Section 7.2.3.4) • What is class ’signal’ for ? (cf. Section 7.2.3.5) • What is class ’core’ for ? (cf. Section 7.2.3.6) • What is class ’description’ for ? (cf. Section 7.2.3.7) • What is class ’achse’ for ? (cf. Section 7.2.3.8) • Why is class ’achse’ called ’achse’ and not ’axis’ ? (cf. Section 7.2.3.9) • What is class ’unit’ for ? (cf. Section 7.2.3.10) • How is class ’unit’ used in TSTOOL ? (cf. Section 7.2.3.11) 4. Extending TSTOOL (cf. Section 7.2.4) 93 • How can I write a script to automatize common tasks ? (cf. Section 7.2.4.1) 5. Miscellaneous questions (cf. Section 7.2.5) • What’s the difference between history and commandlines ? (cf. Section 7.2.5.1) • Is it TSTOOL, TSTool, OpenTSTOOL, OpenTSTool, ... ? (cf. Section 7.2.5.2) 6. Frequently encountered errors (cf. Section 7.2.6) • Using a column vector to create a one-dimensional signal (cf. Section 7.2.6.1) • What does the error message ”Attempt to execute SCRIPT . . . as a function.” mean? (cf. Section 7.2.2.4) 7.2 Answers 7.2.1 Introduction and general information 7.2.1.1 What is TSTOOL ? TSTOOL is a software package for nonlinear time series analysis, though it has a lot of features a general signal analysis package would also have. 7.2.1.2 What software is required to run TSTOOL ? TSTOOL is written in MATLAB, a powerful language for scientific computing, and in C++. Therefore you need MATLAB version 6.5 or higher to run tstool. Unfortunately, MATLAB is not free software! 7.2.1.3 On which systems does TSTOOL run ? TSTOOL should run on all platforms which run MATLAB and for which you can compile the included mex-files, which are functions written in C or C++ that extend MATLAB’s set of build-in functions. We try to ship pre-compiled mex-files with TSTOOL for the most popular operating systems, namely Windows XP (32 and 64 bit), Linux (32 and 64 bit), Mac OS X and Solaris. However, since the compiled mex-files are not compatible between all MATLAB versions are operating systems, you may have to compile the mex-files yourself (see the installation section for details). 7.2.1.4 What about Octave or other Matlab like programming environments ? Octave1 is a freely available language for scientific computing that strongly resembles MATLAB. Unfortunately, Octave is not fully compatible to MATLAB, so TSTOOL does not work with it. TSTOOL makes extended use of the object oriented features of MATLAB. In the current version of Octave (3.0) there’s no full support of classes. Even if classes will be supported in future, it’s not sure wheter TSTOOL will work properly. There are several other Matlab like programming environments, e.g. Mideva2 or Scilab3 . Up to now, it is not possible to use TSTOOL with these packages. 1 See URL http://www.gnu.org/software/octave URL http://www.mathtools.com 3 See URL http://www-rocq.inria.fr/scilab/ 2 See 94 7.2.2 Installation of TSTOOL 7.2.2.1 All lines in the OpenTSTOOL/tstoolbox/mex/*.m are comments, is this right? Yes, this are the comment-texts for the compiled mex functions (e.g. type help amutual at the matlab prompt). 7.2.2.2 Where are the precompiled Mex-Files? They are in OpenTSTOOL/tstoolbox/mex/<MEXEXT>, where <MEXEXT> is the extension for mex files on your system. You can determine this extension by using the command mexext in Matlab. 7.2.2.3 There are more than one file called e.g. amutual.m, why? For some functions there are up to three versions of the file: • OpenTSTOOL/tstoolbox/@signal/amutual.m Function that invokes the underlying mex function. It uses signal objects as output and input. • OpenTSTOOL/tstoolbox/mex/amutual.m Help text for the compiled mex function (e.g. type help amutual on the matlab prompt). • OpenTSTOOL/tstoolbox/mex/mexsol/amutual.mexsol, OpenTSTOOL/tstoolbox/mex/mexsg64/amutual.mexsg64, OpenTSTOOL/tstoolbox/mex/mexglx/amutual.mexglx, OpenTSTOOL/tstoolbox/mex/mexw32/amutual.mexw32 Precompiled mex files for Linux x86, Windows, Mac OS X and Solaris are shipped with TSTOOL. Only one of these files may be present on your system, depending on the version of TSTOOL you have downloaded. 7.2.2.4 What does the error message ”Attempt to execute SCRIPT . . . as a function.” mean? Matlab cannot find the correct mex files for this systems and so it tries to execute the scripts OpenTSTOOL/mex/*.m (which are only the help texts for the mex files). There are many possibilities for this error: • You downloaded the wrong version of TSTOOL. • The path setting made by settspath.m are not correct. Type path at the matlab prompt and look for the path setting for the mex directory (see 7.2.2.3). • The mex files are not present in the directory noted in 7.2.2.3. 7.2.2.5 What does the error message ”This application has failed to start because the application configuration is incorrect.” mean? This happens on Windows systems without the Visual C++ 2008 run-time libraries. Since the mexfiles for Windows are compiled with VC++ 2008, these run-time libraries are needed for the proper execution of the compiled Windows mex-files that ship with TSTOOL. Please see the installation section for details on how to install the run-time libraries. 95 7.2.3 Working with TSTOOL 7.2.3.1 How do I create a signal from my time-series data ? Suppose the time-series data is given as the row vector y. >> s = signal(y) >> view(s) If y is a column vector, the following syntax must be used: >> s = signal(y’) >> view(s) Suppose the data was recorded with a samplerate of 8 kHz : >> s = signal(y, 8000) >> view(s) 7.2.3.2 How do I create a signal with logarithmic spacing ? Suppose you have data vector y whose values were recorded at 3 Hz, 6 Hz, 12 Hz, 24 Hz ... a = achse(unit(’Hz’), 3, 2, ’log’) s = signal(y, a) view(s) 7.2.3.3 How do I create a signal from non-uniformly sampled data ? Suppose you have a data vector y (of length 4) whose values were recorded at 3 Hz, 5 Hz, 8 Hz, 14.5 Hz a = achse(unit(’Hz’), [3 5 8 14.5]) s = signal(y, a) view(s) 7.2.3.4 How do I change the type of plot that I get with view ? The way view plots a signal depends on the attributes of the signal. It is possible to give view a hint which type of plot to prefer. This hint can be set with the command setplothint. The possible plot types can be obtained by issuing help signal/view at the Matlab prompt. However, if the signal does not support the desired type of plotting (e.g. a one-dimensional time-series can not be visualized as orbit), view will use the default plot type for the data. s = signal(rand(1000, 3)); s = setplothint(s, ’3dpoints’); view(s) 96 7.2.3.5 What is class ’signal’ for ? Class signal is TSTOOL’s main class. Objects of this type model real world signals. A signal does not only store the pure sample values, it holds much more information like axes, units of sample values or the axes units, and even more descriptive information like labels, command lines and a processing history. The majority of functions in the tstoolbox take a signal as input argument and return a processed signal as output. This allows for combining or chaining of several processing steps in order to get the desired output. 7.2.3.6 What is class ’core’ for ? Class core is a base class of class signal. An object of type core stores the pure sample values of a signal, without any additional descriptive information. The separation of the numerical and the descriptive part of a signal simplifies the writing of m-files that work on signals. 7.2.3.7 What is class ’description’ for ? Class description is the second base class of class signal. An object of type description stores all descriptive information belonging to a signal. 7.2.3.8 What is class ’achse’ for ? Class achse models an axis, e.g. a time axis or a frequency axis. A signal has a least one axis (if it is a one dimensional signal). A multidimensional signal has several achse objects, one for each dimension. An achse object is basically described by an object of class unit and the spacing values. The spacing may be linear, logarithmic or arbitrary (in case of non-uniform sampling). 7.2.3.9 Why is class ’achse’ called ’achse’ and not ’axis’ ? The names axis and axes are already occupied in Matlab. So, achse, which is the german translation of axis, was used as name for that class. 7.2.3.10 What is class ’unit’ for ? Objects of class ’unit’ try to model physical units. No one wonders that his computer can multiply real or complex numbers. But in physics or engineering, you also have to mulitply or divide physical units, just think of Ohm’s law : R = U/I 7.2.3.11 How is class ’unit’ used in TSTOOL ? Class unit is used as a part of every achse object and as part of a description object. Handling and processing of units is optional for functions that work on signals, because many nonlinear signal analysis functions do not allow consistent handling of units. 7.2.4 Extending TSTOOL Of course it’s possible to extend TSTOOL with some custom functionality or to use parts of TSTOOL in your own m-files, just as with other toolboxes for Matlab. 97 7.2.4.1 How can I write a script to automatize common tasks ? One way to obtain a script is to execute the desired analysis steps with one example signal. The output of this tasks will again be a signal that has stored the syntax of the executed steps in its description. Using the command commandlines(result) will give you this syntax. With copy and paste, it’s possible to create a script file from that output. 7.2.5 Miscellaneous questions 7.2.5.1 What’s the difference between history and commandlines ? Both attributes of class description are used to record the processing history of a signal. But, while history contains a list human readable entries, commandlines stores the exact syntax of the commands that were applied to the signal. 7.2.5.2 Is it TSTOOL, TSTool, OpenTSTOOL, OpenTSTool, ... ? Initially it was called TSTOOL, but since Matlab now has its own tstool command, we changed the Matlab command to opentstool, but keep mostly TSTOOL in the manual and homepage for brevity’s sake. Regarding the spelling, we’re not entirely sure, either. 7.2.6 Frequently encountered errors 7.2.6.1 Using a column vector to create a one-dimensional signal TSTOOL stores one-dimensional signals always as row vectors. Giving a column vector will cause unexpected behaviour with most routines that process signals: >> s = signal(sin(0:0.5:100)) s = signal object Dlens : 1 201 X-Axis 1 : | X-Axis 2 : | Name : Type : Attributes of data values : | Comment : History : 16-Aug-1999 19:15:01 : Imported from MATLAB workspace Instead, a row vector must be given to create the desired one-dimensional signal: 98 >> s = signal(sin(0:0.5:100)’) s = signal object Dlens : 201 X-Axis 1 : | Name : Type : Attributes of data values : | Comment : History : 16-Aug-1999 19:16:58 : Imported from MATLAB workspace 99 Bibliography 101 [1] Ott, E. (1993), Chaos in Dynamical Systems, Cambridge Cambridge, University Press. [2] Thompson, J. M. T., H. B. Stewart (1986), Nonlinear Dynamics and Chaos, Chichester, Wiley. [3] Schuster, H. G. (1988), Deterministic Chaos, 2nd ed., Weinheim, VHC Publishers. [4] Berg´e, P., Y. Pomeau, C. Vidal (1984), Order within Chaos: Towards a Deterministic Approach to Turbulence, New York, John Wiley and Sons. [5] Moon, F. C. (1992), Chaotic and Fractal Dynamics, New York, John Wiley and Sons. [6] Gaponov-Grekhov, A. V., M. I. Rabinovich, (1992), Nonlinearities in Action – Oscillations, Chaos, Order, Fractals, Berlin, Springer. [7] Lauterborn, W., J. Holzfuss, (1991) , “Acoustic chaos”, Int. J. Bifurcation and Chaos, 1, pp.13-26. [8] Lauterborn, W., T. Kurz, U. Parlitz, (1997), “Experimental Nonlinear Physics”, Int. J. Bifurcation and Chaos, 7, pp.2003-2033. [9] Lauterborn, W., U. Parlitz, (1988), “Methods of chaos physics and their application to acoustics”, J. Acoust. Soc. Am., 84, pp.1975-1993. [10] Packard, N.H., J.P. Crutchfield, J.D. Farmer, R.S. Shaw (1980), “Geometry from a time series”, Phys. Rev. Lett., 45, pp.712-716. [11] Takens, F. (1981), “Detecting strange attractors in turbulence”, in Dynamical Systems and Turbulence, eds. Rand, D.A. & Young, L.-S. , Berlin, Springer, pp.366-381. [12] Kantz, H., & T. Schreiber (1997), Nonlinear Time Series Analysis, Cambridge University Press, Cambridge. [13] Abarbanel, H.D.I. (1996), Analysis of Observed Chaotic Data, Springer, New York. [14] Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., & Tsimring, L.S. (1993), “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys., 65(4), pp.1331-1392. [15] Grassberger, P., Schreiber, T. & Schaffrath, C. (1991), “Nonlinear time sequence analysis,” Int. J. Bif. Chaos, 1(3), pp.521-547. [16] Sauer, T., Y. Yorke, M. Casdagli (1991), “Embedology”, J. Stat. Phys., 65, pp.579-616. [17] Sauer, T., J.A. Yorke (1993), “How many delay coordinates do you need ?”, Int. J. Bifurcation and Chaos, 3, pp.737-744. [18] Casdagli, M., S. Eubank, J.D. Farmer, J. Gibson (1991), “State space reconstruction in the presence of noise”, Physica D, 51, pp.52-98. [19] Gibson, J.F., J.D. Farmer, M. Casdagli, S. Eubank (1992), “An analytic approach to practical state space reconstruction” Physica D, 57, pp.1-30. [20] Broomhead, D.S., G.P. King (1986), “Extracting qualitative dynamics from experimental data”, Physica D, 20, pp.217-236. [21] Landa, P.S., M.G. Rosenblum (1991), “Time series analysis for system identification and diagnostics”, Physica D, 48, pp.232-254. [22] Palus, M., I. Dvorak (1992), “Singular-value decomposition in attractor reconstruction: pitfalls and precautions”, Physica D, 55, pp.221-234. [23] Sauer, T.(1994), “Reconstruction of dynamical systems from interspike intervals”, Phys. Rev. Lett., 72, pp.3811-3814. [24] Castro, R., T. Sauer (1997), “Correlation dimension of attractors through interspike intervals”, Phys. Rev. E, 55(1), pp.287-290. 102 [25] Racicot, D.M., A. Longtin (1997), “Interspike interval attractors from chaotically driven neuron models”, Physica D, 104, pp.184-204. [26] Stark, J., D.S. Broomhead, M.E. Davies, J. Huke (1996), “Takens embedding theorems for forced and stochastic systems”, in: Proceedings of the 2nd World Congress of Nonlinear Analysts, Athens, greece, July 1996. [27] Kennedy, M.P. (1994), “Chaos in the Colpitts oscilator”, IEEE Trans. Circuits Syst., 41(11), pp.771-774. [28] Cenys, A., K. Pyragas (1988), “Estimation of the number of degrees of freedom from chaotic time series”, Phys. Lett. A, 129, pp.227-230. [29] Buzug, Th., T. Reimers, G. Pfister (1990), “Optimal reconstruction of strange Attractors from purely geometrical arguments”, Europhys. Lett., 13, pp.605-610. [30] Alecsi´c, Z. (1991), “Estimating the embedding dimension”, Physica D, 52, pp.362-368. [31] Buzug, Th., G. Pfister (1992), “Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors”, Phys. Rev. A, 45, pp.7073-7084. [32] Gao, J., Z. Zheng (1993), “Local exponential divergence plot and optimal embedding of a chaotic time series”, Phys. Lett. A, 181, pp.153-158. [33] Gao, J., Z. Zheng (1994). “Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time series”, Phys. Rev. E, 49, pp.3807-3814. [34] Huerta, R., C. Santa Cruz, J.R. Dorronsore, V. L`opez (1995), “Local state-space reconstruction using averaged scalar products of dynamical-system flow vectors”, Europhys. Lett., 29, pp.13-18. [35] Liebert, W., K. Pawelzik, H.G. Schuster (1991), “Optimal embeddings of chaotic attractors from topological considerations”, Europhys. Lett., 14, pp.521-526. [36] Kennel, M.B., R. Brown, H.D.I. Abarbanel (1992), “Determining embedding dimension for phasespace reconstruction using a geometrical construction”, Phys. Rev. A, 45, pp.3403-3411. [37] Fredkin, D.R., J.A. Rice (1995), “Method of false nearest neigbors: a cautionary note”, Phys. Rev. E, 51(4), pp. 2950-2954. [38] Cao, L. (1997), “Practical method for determining the minimum embedding dimension of a scalar time series”, Physcai D, 110, pp. 43-50. [39] Kember, G., A.C. Fowler (1993), “A correlation function for choosing time delays in phase portrait reconstructions”, Phys. Lett. A, 179, pp.72-80. [40] Rosenstein, M.T., J.J. Collins, C.J. De Luca (1994), “Reconstruction expansion as a geometrybased framework for choosing proper delay times”, Physica D, 73, pp.82-98. [41] Frazer, A.M., H.L. Swinney (1986), “Independent coordinates in strange attractors from mutual information”, Phys. Rev. A, 33, pp.1134-1140. [42] Frazer, A.M. (1989), “Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria”, Physica D, 34, pp.391-404. [43] Frazer, A.M. (1989), “Information and entropy in strange attractors”, IEEE Trans. Info. Theory, 35, pp.245-262. [44] Liebert, W., H.G. Schuster (1989), “Proper choice of the time delay for the analysis of chaotic time series”, Phys. Lett. A, 142, pp.107-111. [45] Martinerie, J.M., A.M. Albano, A.I. Mees, P.E. Rapp (1992), “Mutual information, strange attractors, and the optimal estimation of dimension”, Phys. Rev. A, 45, pp.7058-7064. 103 [46] Broomhead, D.S., J.P. Huke, M.R. Muldoon (1992), “Linear filters and nonlinear systems”, J. Roy. Stat. Soc., B54, pp.373-382. [47] Davies, M.E., & K.M. Campbell (1996), “Linear recursive filters and nonlinear dynamics” Nonlinearity, 9, pp.487-499. [48] Kaplan, D., T. Schreiber (1996), “Signal separation by nonlinear projections: The fetal electrocardiogram”, Phys. Rev. E, 53(5), pp.R4326-R4329. [49] Grassberger, P., R. Hegger, H. Kantz, C. Schaffrath, T. Schreiber (1993), “On noise reduction methods for chaotic data” CHAOS, 3, pp.127-141. [50] Kantz, H., T. Schreiber, I. Hoffmann, T. Buzug, G. Pfister, C.G. Flepp, J. Simonet, R. Badii, E. Brun (1993), “Nonlinear noise reduction: A case study on experimental data”, Phys. Rev. E, 48, pp.1529-1538. [51] Kostelich, E.J.,T. Schreiber (1993), “Noise reduction in chaotic time-series data: A survey of common methods”, Phys. Rev. E, 48, pp.1752-1763. [52] Theiler, J., B. Galdrikian, A. Longtin, S. Eubank, J.D. Farmer (1992), “Using surrogate data to detect nonlinearity in time series” in: Nonlinear Modeling and Forecasting, eds. M. Casdagli and S. Eubank, SFI Studies in the Sciences of Complexity, Vol.XII (Reading, MA,Addison-Wesley), pp.163-188. [53] Theiler, J., S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer (1992), “Testing for nonlinearity in time series: the method of surrogate data”, Physica D, 58, pp.77-94. [54] Provenzale, A., L.A. Smith, R. Vio, G. Murante (1992), “Distiguishing between low-dimensional dynamics and randomness in measured time series”, Physica D, 58, pp.31-49. [55] Smith, L. (1992), “Identification and prediction of low dimensional dynamics”, Physica D, 58, pp.50-76. [56] Takens, F. (1993), “Detecting nonlinearities in stationary time series”, Int. J. of Bifurcation and Chaos, 3, pp.241-256. [57] Wayland, R., D. Bromley, D. Pickett, A. Passamante (1993), “Recognizing determinism in a time series”, Phys. Rev. Lett., 70, pp.580-582. [58] Palus, M., V. Albrecht,I. Dvorak (1993), “Information theoretic test for nonlinearity in time series”, Phys. Lett. A, 175, pp.203-209. [59] Kaplan, D. (1994), “Exceptional events as evidence for determinism”, Physica D, 73, pp.38-48. [60] Salvino, L.W., R. Cawley (1994), “Smoothness implies determinism: a method to detect it in time series”, Phys. Rev. Lett., 73, pp.1091-1094. [61] Savit, R.M. Green (1991), “Time series and dependent variables”, Physica D, 50, pp.95-116. [62] Rapp, P.E., A.M. Albano, I.D. Zimmerman, M.A. Jim´enez-Molta˜ no (1994), “Phase-randomized surrogates can produce spurious identifications of non-random structure”, Phys. Lett. A, 192, pp.27-33. [63] Theiler, J. (1995), “On the evidence for low-dimensional chaos in an epileptic electroencephalogram”, Phys. Lett. A, 196, pp.335-341. [64] Schreiber, T., A. Schmitz (1996), “Improved surrogate data for nonlinearity tests”, Phys. Rev. Lett., 77(4), pp.635-638. [65] Schreiber. T (1998), “Constrained randomization of time series data”, Phys. Rev. Lett., 80(10), pp.2105-2108. [66] Judd, K., A. Mees (1995), “On selecting models for nonlinear time series”, Physica D, 82, pp. 426-444. 104 [67] Aguirre, L.A., S.A. Billings (1995), “Identification of models for chaotic systems from noisy data: implications for performance and nonlinear filtering”, Physica D, 85, pp. 239-258. [68] Aguirre, L.A., E.M.A.M. Mendes (1996), “Global nonlinear polynomial models: structure, term clustering and fixed points”, Int. J. Bifurc. Chaos, 6(2), pp.279-294. [69] Allie, S., A. Mees, K. Judd, D. Watson (1997), “Reconstructing noisy dynamical systems by triangulation”, Phys. Rev. E, 55(1), pp. 87-93. [70] Szpiro, G.G. (1997), “Forecasting chaotic time series with genetic algorithms”, Phys. Rev. E, 55(3), pp. 2557-2568. [71] Jaeger, L., H. Kantz (1997), “Effective deterministic models for chaotic dynamics perturbed by noise”, Phys. Rev. E, 55(5), pp. 5234-5247. [72] Farmer, J.D., J.J. Sidorowich (1987), “Predicting chaotic time series”, Phys. Rev. Lett., 59, pp.845-848. [73] Casdagli, M. (1989), “Nonlinear Prediction of chaotic time series”, Physica D, 35, pp.335-356. [74] Brown, R. N.F. Rulkov, E.R. Tracy (1994), “Modeling and synchronizing chaotic systems from time-series data”, Phys. Rev. E, 49, pp.3784-3800. [75] Grassberger, P., I. Procaccia (1983), “On the characterization of strange attractors”, Phys. Rev. Lett., 50, pp.346-349. [76] Theiler, J. (1986), “Spurious dimension from correlation algorithms applied to limited time-series data”, Phys. Rev. A, 34, pp.2427-2431. [77] Badii, R., A. Politi (1984), “Hausdorff dimension and uniformity factor of strange attractors”, Phys. Rev. Lett, 52, pp.1661-1664. [78] Badii, R., A. Politi (1985), “Statistical description of chaotic attractors”, J. Stat. Phys., 40, pp.725-750. [79] Grassberger, P. (1985), “Generalizations of the Hausdorff dimension of fractal measures”, Phys. Lett. A, 107, pp.101-105. [80] Schreiber, T. (1995), “Efficient neighbor searching in nonlinear times series analysis”, Int. J. Bifurcation and Chaos, 5, pp.349-358. [81] Holzfuss, J., G. Mayer-Kress (1986), “An approach to error-estimation in the application of dimension algorithms”, in [82], pp.114-122. [82] Mayer-Kress, G. (ed.) (1986), Dimensions and Entropies in Chaotic Systems – Quantification of Complex Behavior, Berlin, Springer. [83] Theiler, J. (1990), “Estimating fractal dimension”, J. Opt. Soc. Am. A, 7, pp.1055-1073. [84] Broggi, G. (1988), “Evaluation of dimensions and entropies of chaotic systems”, J. Opt. Soc. Am. B, 5, pp.1020-1028. [85] Oseledec, V.I. (1968), “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems”, Trans. Moscow Math. Soc., 19, pp.197-231. [86] Benettin, G., L. Galgani, A. Giorgilli, J.-M. Strelcyn (1980), “Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part II: Numerical application ” Meccanica, 15, pp.21-30. [87] Shimada, I., T. Nagashima (1979), “A numerical approach to ergodic problems of dissipative dynamical systems”, Prog. Theor. Phys., 61, pp.1605-1616. [88] Eckmann, J.-P., D. Ruelle (1985), “Ergodic theory of chaos and strange attractors”, Rev. Mod. Phys., 57, pp.617-656. 105 [89] Geist, K., U. Parlitz, W. Lauterborn (1990), “Comparison of Different Methods for Computing Lyapunov Exponents”, Prog. Theor. Phys., 83, pp.875-893. [90] Wolf, A., J.B. Swift, L. Swinney, J.A. Vastano (1985), “Determining Lyapunov exponents from a time series”, Physica D, 16, pp.285-317. [91] Sano, M., Y. Sawada (1985), “Measurement of the Lyapunov spectrum from a chaotic time series”, Phys. Rev. Lett., 55, pp.1082-1085. [92] Eckmann, J.-P., S.O. Kamphorst, D. Ruelle, S. Ciliberto (1986), “Lyapunov exponents from time series”, Phys. Rev. A, 34, pp.4971-4979. [93] Stoop, R., P.F. Meier (1988), “Evaluation of Lyapunov exponents and scaling functions from time series”, J. Opt. Soc. Am. B, 5, pp.1037-1045. [94] Holzfuss, J., W. Lauterborn (1989), “Liapunov exponents from a time series of acoustic chaos”, Phys. Rev. A, 39, pp.2146-2152. [95] Stoop, R., J. Parisi (1991), “Calculation of Lyapunov exponents avoiding spurious elements”, Physica D, 50, pp.89-94. [96] Zeng, X., R. Eykholt, R.A. Pielke (1991), “Estimating the Lyapunov-exponent spectrum from short time series of low precision”, Phys. Rev. Lett., 66, pp.3229-3232. [97] Zeng, X., R.A. Pielke, R. Eykholt (1992), “Extracting Lyapunov exponents from short time series of low precision”, Modern Phys. Lett. B, 6, pp.55-75. [98] Parlitz, U. (1993), “Lyapunov exponents from Chua’s circuit”, J. Circuits, Systems and Computers, 3,pp.507-523. [99] Kruel, Th.M., M. Eiswirth, F.W. Schneider (1993), “Computation of Lyapunov spectra: Effect of interactive noise and application to a chemical oscillator”, Physica D, 63, pp.117-137. [100] Briggs, K. (1990), “An improved method for estimating Liapunov exponents of chaotic time series”, Phys. Lett. A, 151, pp.27-32. [101] Bryant, P., R. Brown, H.D.I. Abarbanel (1990), “Lyapunov exponents from observed time series”, Phys. Rev. Lett., 65, pp.1523-1526. [102] Brown, R., P. Bryant, H.D.I. Abarbanel (1991), “Computing the Lyapunov spectrum of a dynamical system from an observed time series”, Phys. Rev. A, 43, pp.2787-2806. [103] Abarbanel, H.D.I., R. Brown, M.B. Kennel (1991), “Lyapunov exponents in chaotic systems: their importance and their evaluation using observed data”, Int. J. Mod. Phys. B, 5, pp.13471375. [104] Holzfuss, J, U. Parlitz (1991), “Lyapunov exponents from time series”, Proceedings of the Conference Lyapunov Exponents, Oberwolfach 1990, eds. L. Arnold, H. Crauel, J.-P. Eckmann, in: Lecture Notes in Mathematics, Springer Verlag. [105] Parlitz, U. (1992), “Identification of true and spurios Lyapunov exponents from time series”, Int. J. Bifurcation and Chaos, 2, pp.155-165. [106] Kadtke, J.B., J. Brush, J. Holzfuss (1993), “Global dynamical equations and Lyapunov exponents from noisy chaotic time series”, Int. J. Bifurcation Chaos, 3, pp.607-616. [107] Gencay, R., W.D. Dechert (1992), “An algorithm for the n Lyapunov exponents of an ndimensional unknown dynamical system”, Physica D, 59, pp.142-157. [108] Eckmann, J.-P., D. Ruelle (1992), “Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems”, Physica D, 56, pp.185-187. 106 [109] Ellner, S., A.R. Gallant, D. McCaffrey, D. Nychka (1991), “Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data”, Phys. Lett. A, 153,pp.357363. [110] Fell, J., J. R¨ oschke, P. Beckmann (1993), “Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep”, Biol. Cybern., 69, pp.139-146. [111] Fell, J., P. Beckmann (1994), “Resonance-like phenomena in Lyapunov calculations from data reconstructed by the time-delay method” Phys. Lett. A, 190, pp.172-176. [112] Sato, S., M. Sano, Y. Sawada (1987), “Practical methods of measuring the generalized dimension and largest Lyapunov exponent in high dimensional chaotic systems”, Prog. Theor. Phys., 77, pp.1-5. [113] Kurths, J., H. Herzel (1987), “An attractor in solar time series”, Physica D, 25, pp.165-172. [114] D¨ ammig, M., F. Mitschke (1993), “Estimation of Lyapunov exponents from time series: the stochastic case”, Phys. Lett. A, 178, pp.385-394. [115] Rosenstein, M.T., J.J. Collins, C.J. de Luca (1993), “A practical method for calculating largest Lyapunov exponents from small data sets”, Physica D, 65, pp.117. [116] Kantz, H. (1994), “A robust method to estimate the maximal Lyapunov exponent of a time series”, Phys. Lett. A, 185, pp.77-87. [117] Fujisaka, H.,T. Yamada (1993), “Stability theory of synchronized motion in coupled-oscillator systems,” Prog. Theor. Phys., 69, pp.32-46. [118] Singer, W. (1993), “Synchronization of cortical activity and its putative role in information processing and learning,” Annu. Rev. Physiol., 55, pp.349-374. [119] Ashwin, P., J. Buescu, I. Stewart (1994), “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A, 193,pp.126-139. [120] Heagy, J.F., T.L. Carroll, L.M. Pecora (1994), “Synchronous chaos in coupled oscillator systems,” Phys. Rev. E, 50, pp.1874-1885. [121] Lai, Y.-C., C. Grebogi (1994), “Synchronization of spatiotemporal chaotic systems by feedback control,” Phys. Rev. E, 50,pp.1894-1899. [122] Collins, J.J.,I. Stewart (1994), “A group-theoretic approach to rings of coupled biological oscillators,” Biol. Cybern., 71, pp.95-103. [123] Lindner, J.F., B.K. Meadows, W.L. Ditto, M.E. Inchiosa, A.R. Bulsara (1995), “Array enhanced stochastic resonance and spatiotemporal synchronization,” Phys. Rev. Lett., 75, pp.3-6. [124] Braiman, Y., W.L. Ditto, K. Wiesenfeld, M.L. Spano (1995), “Disorder-enhanced synchronization,” Phys. Lett. A 206, pp.54-60; Braiman, Y., J.F. Lindner, W.L. Ditto (1995), “Taming spatiotemporal chaos with disorder,” Nature, 378, pp.465-467. [125] Pecora, L.M., T.L. Carroll (1990), “Synchronization in chaotic systems,” Phys. Rev. Lett., 64, pp.821-824. [126] Brown, R., N.F. Rulkov, E.R. Tracy (1994), “Modelling and synchronizing chaotic systems from experimental data,” Phys. Lett. A, 194, pp.71-76. [127] Kocarev. L., U. Parlitz (1995), “General approach for chaotic synchronization with applications to communication”, Phys. Rev. Lett., 74(25), pp.5028-5031. [128] Parlitz, U., L. Junge, L. Kocarev, (1996), “Synchronization based parameter estimation from time series”, Phys. Rev. E, 54, pp.6253-6529. 107 [129] Parlitz, U., L. Kocarev, T. Stojanovski, H. Preckel (1996), “Encoding messages using chaotic synchronization,” Phys. Rev. E, 53(5), pp.4351-4361. [130] Rulkov, N.F., K.M. Sushchik, L.S. Tsimring, H.D.I. Abarbanel, (1995), “Generalized synchronization of chaos in directionally coupled chaotic systems,” Phys. Rev. E, 51, pp.980-994. [131] Kocarev, L., U. Parlitz (1996), “Generalized synchronization, predictability and equivalence of uni-directionally coupled dynamical systems”, Phys. Rev. Lett., 76(11), pp.1816-1819. [132] Abarbanel, H.D.I., N.F. Rulkov, M.M. Sushchik (1996), “Generalized synchronization of chaos: The auxiliary system approach”, Phys. Rev. E, 53(5), pp.4528-4535. [133] Parlitz, U., L. Junge, L. Kocarev (1997), “Subharmonic entrainment of unstable period orbits and generalized synchronization”, Phys. Rev. Lett., 79(17), pp.3158. [134] Parlitz, U., L. Kocarev (1998), “Synchronization of chaotic systems”, in: “Control of Chaos” Handbook (Ed. H.-G. Schuster), WILEY-VCH. [135] Sirovich, L. (1989), ‘Chaotic dynamics of coherent structures”, Physica D, 37, pp. 126-145. [136] Rico-Martinez, R., K. Krischer, I.G. Kevrekidis, M.C. Kube, J.L. Hudson, (1992), “Discrete - vs. continuous-time nonlinear signal processing of Cu electrodissolution data,” Chem. Eng. Comm. 118, pp.25-48. [137] Parlitz, U. & G. Mayer-Kress (1995), “Predicting low-dimensional spatiotemporal dynamics using discrete wavelet transforms”, Phys. Rev. E, 51(4), pp.R2709-R2711. [138] H.D.I. Abarbanel, Analysis York/Berlin/Heidelberg, 1996); of Observed Chaotic Data, (Springer Verlag, New- [139] S. Arya, D.M. Mount, N.S. Netanyahu, R. Silverman and A. Wu An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions,, (Proc. of the Fifth Annual ACM– SIAM Symp. on Discrete Algorithms, 1994, pp. 573-582) [140] A. Belussi and C. Faloutsos Estimating the Selectivity of Spatial Queries Using the ‘Correlation’ Fractal Dimension, (Conference Proceedings of VLDB, Zurich, Switzerland, Sept. 1995, pp. 299310) [141] S. Berchtold, C. B¨ ohm, D.A. Keim and H.P. Kriegl A cost model for nearest neighbor search in high-dimensional data space, (PODS’97, Tuscon, AZ, pp. 78-86) [142] P. Grassberger, R. Hegger, H. Kantz, C. Schaffrath and T. Schreiber On noise reduction methods for chaotic data, (Chaos, Vol. 3, Nr. 2, 1993, pp. 127-141) [143] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, and B.I. Shraiman Fractal measures and their singularities: The characterization of strange sets, (Phys. Rev. A Vol. 33, Nr. 2, 1986, pp. 1141-1151) [144] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, (Cambridge UP, Cambridge, 1997). [145] J. McNames A Nearest Trajectory Strategy for Time Series Prediction, (Proc. of the International Workshop on Advanced Black-box Techniques for Nonlinear Modeling, 1998, pp. 112-128) [146] U. Parlitz Nonlinear Time-Series Analysis, in (Nonlinear Modeling - Advanced Black-Box Techniques Eds. J.A.K. Suykens and J. Vandewalle Kluwer Academic Publishers, 1998, pp. 209-239) [147] V. Pestov On the geometry of similarity search : dimensionality curse and contraction of measure,, (Maths and comp. science research report, 99-02, VUW, January 1999, pp. 7), submitted for publication [148] T. Schreiber Efficient neighbor searching in nonlinear time-series analysis,, (Int. J. Bifurcation and Chaos, 5, pp. 349-358) 108 [149] R. Sedgewick Algorithms in C++, Third Edition, (Addison-Wesley, 1998) [150] W. van de Water and P. Schram Generalized dimensions from near-neighbor information, (Phys. Rev. A, Vol. 37, Nr. 8, 1988, pp. 3318-3125) [151] L. F. Shampine and M. K. Gordon, “The inital value problem” 109

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement